He was a member of the Bellman's Crew, which was of course led by Bellman. This expedition hunted Snarks. However, when he apparently found a Snark (which was a Boojum), he disappeared. The expedition was ended after that.
The Baker was created by Lewis Carroll in his 1874 nonsense-poem "The Hunting of the Snark". He was illustrated by Henry Holiday.
In 2002, Alan Moore took up the crew in the second volume of League of Extraordinary Gentlemen. Instead of hunting for Snarks, they were an expedition who ventured down the rabbit hole. The Baker also disappeared in this version.
In 2011, Roger Langridge created a comic called Snarked, which was based on Carroll's original poem. In the #0 issue, Henry Holiday's original illustrations were reprinted along with Carrol's poem and the Baker appeared in a few of those illustrations. In issue #1, Langridge retold Hunting of the Snark in his own words providing another appearance for the Baker.
Major Story Arcs
Bellman Expedition (League of Extraordinary Gentlemen)
The Bonnet Maker was a part of the Bellman Expedition, which went down the "Rabbit Hole" like Miss A.L. He, unlike the rest of the crew, did not return.
After the Baker disappeared, the rest of the crew returned home. They were considered the laughing stock of the town because no one believed their story. Even the Baker's brother, the Barber, had his doubts.
The Bellman's Crew was persauded by Wilburforce J. Walrus, who is a conman, that their friend the Baker had not disappeared forever but merely was sent forward in time. Believing his tall tale, the crew decided to take the Walrus and the Carpenter and royals to Snark Island.