Dr. Baxter Stockman was a sociopath scientist that developed killer robots called Mousers that he announced to the press would solve New York city's vermin problem. In truth the robots had an ulterior purpose of burrowing under the city and stealing the money from bank vaults. When his computer programmer April O'Neil started to grow suspicious of the Mouser's true purpose Stockman led her to an underground factory were he was developing hundreds of Mousers. When she questioned why he would do this as he could make millions legally, he answered that money is only an incidental by product. He was really doing it "because it's FUN". When April tried to get away Stockman opened a trap door to sewer level and set the Mousers on her.
Stockman stepped up his plans when he sent a video of himself surrounded in shadow saying that his intention was to extort money from the city but holding buildings hostages, if a ransome was not delivered via his complex method he would destroy the building in question. To show he was not joking he used his Mousers to destroy the Retxab Building, claiming that one of the World Trade Center towers would be next if he did not recieve twenty million dollars. His plan was thwarted when April unexpectedly turned up alive with her rescuers, the Teenage Mutant Ninja Turtles to stop him. The four turtles easily managed to subdue Stockman but not before he initiated a program that recalled the Mousers to the lab were they would self destruct on arrival and sealed all the exits. April and Donatello managed to narrowly abort the self destruct.
Years later Stockman managed to break into a DARPA facilty, killing several guards and sealing himself in one of the labs. When security managed to get into the room they found the dead bodies of the guards but also the body of Baxter Stockman that had his head cut open and brain removed. Stockman had transferred his mind into that of a robot and started making its way to New York to exact revenge on the person he blamed for his misfortunes: April O'Neil.
Reaching the city he managed to track down O'Neil as she was driving to the grocery store and attacked her by bursting out of the street below and ripping her car open. He stabbed her in the shoulder with the tip of a syringe and vows this is only the beginning of the pain he will bring upon her, with that he leaves via the sewers. The turtles and their ally Nobody manage to catch up with Stockman and a massive rooftop battle ensues. Stockman has the upper hand until the battle draws too much attention he chooses to retreat, jumping off the roof and grabbing on to a police helicopter to slow his fall he sustains damage and retreats to the sewers were he blacks out. When he wakes he's shocked to find that the robot body is repairing itself, even growing a new arm to replace the one that was blown off in the fight.
The turtles and Casey Jones once again track down Stockman in the sewers. Stockman drops a live cable hoping that when it reaches the water it would electrocute his enemies. Quick thinking by Leonardo has the Baxter be the one that's electrocuted until he explodes into pieces. The only thing left of Stockman is his glasses which were inside the machine.
When the Utroms make first contact with humanity and set up a base in the Atlantic Ocean the turtles arrange for the human friends April, Casey and Shadow to visit the base. When stepping through the security scanner it detects nanobots in April's body. Under medical examination it is revealed that the nanobots are of Utrom design, the Utroms relate a story that during their first sojourn on Earth they would occasionally hire humans to help with their work though they would not reveal the their true nature to them. One of these humans was Baxter Stockman who stole a batch of nanbots. Obviously not wanted to subject themselves to public scrutiny, they did not pursue him. It is revealed that this is what Stockman had injected her with when he stabbed her shoulder years prior.
Donatello goes to a secret lab in the sewers were it is revealed that unbeknownst to anyone he had kept what was left of Stockman captive for years. It had been four years, twelve days, six hours and forty-three minutes since they had talked last. When Don responds that he's been busy the Baxterbot taunts him "probably off battling the Toe Clan...'fight with honor' or some such nonsense." He does, however, point out that with years alone in the dark he has thought of new solutions to old problems and even claimed to know how to build a time machine.
Donatello tries for hours to get Stockman to reveal the purpose of the nanobots and how to remove them but in the end he gives up. The turtle reveals that his brothers would be appalled to know he had kept Stockman alive bu he did so, hoping there was something of value that could be learned from him. However, Donatello admits he knows Stockman is irredeemably evil and should have been eliminated long ago. Taking out a blaster, he fires at Stockman finally putting him out of his misery with the parting words "Goodbye Baxter. Rot in Hell."
1987 Animated series
Baxter Stockman was a struggling inventor who was unsuccessfully trying to sell his Mouser robots to pest extermination companies but was met with rejection due to the Mouser's being too effective which would eliminate the city's rat problem and would put them out of business. He was approached by the Shredder with an offer to mass produce his Mousers. Unbeknownst to Stockman the ninja planned to use them to hunt down his enemy Splinter. He was hunted down by the Teenage Mutant Ninja Turtles who used Stockman's remote to override the Mouser's when they attacked by the robots. Stockman was blamed for the havoc caused by his creation and arrested. The following episodes had the turtles use Stockman's building as a secondary base. Using Stockman's van and his equipment Donatello made the Turtle Van and later the the Turtle Blimp.
In the show's second season Baxter is broken out of a mental institution by the Shredder. This time Stockman works with the Shredder with full knowledge of what he's involved in and uses his scientific expertis to aid the ninja, particularly in tracking down the thre fragments of the Eyes of Sarnath. Tired of being abused by the Shredder when the three pieces are united he betrays his master and takes the Eyes for himself but is soon put in his place by Saki who manages to take back the Eyes.
Finaly sick of Baxter's failure the Shredder contacts his alien benefactor Krang to send his two hench mutants Bebop and Rocksteady from Dimension-X. Because of a magnetic storm Bebop and Rocksteady could not be sent thought the dimensional portal without someone being sent back. Shredder throws Stockman through the portal in exchange for the two mutants. With no use for the human scientist Krang has Stockman sent to the Disintegrator Unit to dispose of him, unbeknowst to him a fly got into the unit with him and turned Stockman into a mutant fly. Managing to escape from the Disintegrator and opens a portal and returns to Earth. Stockman tracks down the Shredder to take revenge but in his current state his mind is easily influenced and the ninja convinces him that it's the turtles who are his enemies. Still retaining some of his intelligence, he helped set a trap for the turtles by building a device that would render them out of phase with normal time but in the battle Baxter was caught in the machine himself.
Stockman spent months out of phase before using a lightning rod to return to normal and sought revenge on both Shredder and the turtles. Shredder was able to sweet talk Stockman once again by promising that he would have Krang return him to normal if they destroyed the turtles together. The partnership fell though after they were defeated by the turtles and Stockman realised Shredder would never hold up his end of the bargain. Once again he left swearing vengeance on the his former master.
Living in the sewers Stockman came upon an acheological dig. After scaring off the expedition he realised the buried ruins they were researching was actualy an alien ship that had been there for three-hundred years. Stockman befriended the artificial intelligence that ran the ship's computer. Using parts of a pan dimensional warp drive as bait for Shredder. The turtles also got involved and Baxter chose to retreat with the ship but when a piece was missing from the warp drive the ship fell apart in the space between dimenions and Stockman was stuck in limbo.
In limbo Stockman would remain friends with what was left of the ship's computer and after a while helped him escape. His time trapped in limbo had obviously taken a toll as his mind had detoriorated even more, he now had to rely heavily on the computer to keep reminding him who his enemies were and why he even wanted revenge on them, he also displayed more fly-like behavior such as being fascinated by bright lights and sugar. He managed to capture the turtles and imprison them in a clock tower. Knowing that Shredder would want to be the one to dispose of the turtles he decided to use them as bait. The plan of course went awry as the turtles escaped. Baxter tried to follow Shredder through the portal to Dimension-X but was cut off half way and was once again trapped in limbo.
He would return to Earth again without the computer but this time with a new ability to communicate with other flies. He planned to lead an army of them to gain revenge on the Shredder. By this time the Shredder's base the Technodrome was no longer in Dimension-X but rather the Arctic. The cold climate was inhospitable to Stockman and his fly army and so he was easily defeated. Fortunately Krang was intrigued by Stockman's new power and offered Stockman the opportunity to become human again if Stockman and his army would help them take over New York City. Unsuprisingly the plan was thwarted by the the turtles when Donatello used his portable portal to strand the Stockman in another dimension.
Stockman's final appearance on the show had him once again with the computer with a plan for vengeance against the entire world. Using special mutagen he planned to turn the world into insect mutants. Capturing the turtles, they questioned him as to what exactly they had even done to him to warrent revenge with Baxter admitting he can no longer remember...but they'd pay for it anyway. The turtles used Shredder's retro mutagen ray to turn people back to normal and in the process the computer was destroyed and Stockman was once again stuck in limbo presumebly forever.
2003 Animated series
Baxter Stockman owns his own company called Stocktronics were he manufactures his Mousers with the intention of stealing from vaults but publicly claiming his invention is to help with the city's rat problem. His lab tech April O'Neil grows suspicious and when finding out the true purpose of his invention tries to escape through the sewers were she is rescued by the turtles. She also returns with them to stop Stockman.
Stockman isn't merely a solo agent, he was working with the Foot Clan who were financing his invention. After his defeat at the hands of the turtles he worked directly for the Shredder as his chief scientist and built inventions for the Foot to use. After each defeat however he would given "punishments" and each time he was seen his physical condition would grow worse. Over the course of the first season he would be shown to lose an eye, a hand and be bound to a wheel chair after presumably being paralyzed. After the turtles and their allies were seemingly killed during the Foot's raid on April O'Neil's apartment and the Second Time Around the Shredder demanding proof that they had been killed. His tests however came back inconclusive, rather than being the bearer of bad news and risking one final punishment he faked results which showed the Shredder that his enemies had been destroyed.
In truth Casey Jones had taken the turtles, Splinter and April to his Grandmother's old Northampton farmhouse to recover. Stockman used this time to build a giant robot exoskeleton to house his body. In the adaption of "Return to New York" during the turtles attack on the Foot Head Quarters he made his move by attacking the turtles and the Shredder as revenge for all that had happened to him. He was easily defeated by both and as punishment Shredder stripped him of his physical body and had his head attached to a small spider-like bod, this body also had a chip which would make Stockman follow Shredder's every vocal command. Realizing this, Donatello instructed Honycutt to emulate Shredder's voice, recieving conflicting commands the chip overloaded and Stockman was free of Shredder's control.
When Stockman showed up again he had struck up a partnership with the mutant Leatherhead. The mutant built Stockman a mechanical body in exchange for Stockman's help in building a transmat so he could rejoin the Utroms. However when the turtles exposed that Stockman Leatherhead turned against his ally.
2012 Animated series
Baxter Stockman appears as a recurring villain in the Nickelodeon animated series. One of the funny running gags in this show is the Turtles making fun of Baxter as a nontreat. One way they do this is by constantly getting him name wrong calling him things like Dexter Speckman and Boxer Stockboy. He is portrayed by Phil LaMaar.