Bunny was the name given to a bacteria colony born to another universe. She along with the rest of her kind were involved in a war that they lost in a civil action. The one that became known as Bunny was captured as a spoil of war by the Elite who later transformed into their flagship floating fortress and base for operations, taking her away from her home dimension. She not only served as transport but also as a communications satellite, broadcasting the actions of the Elite among the universe via floating cameras and also as prison, keeping inside her a population of dangerous inmates.
Bunny wasn't happy helping the Elite, specialy their leader, Manchester Black and when Superman confront them, Bunny was glad to help him, promising to keep the Elite in check.
However when Black started his campaing against Superman, he freed the prisioners inside Bunny, the Cyborg Superman among them, who take control of the ship and used her to attack Kal-El. Superman could control the ship, pushing her against the ground and incapacitating the Cyborg Superman.
As the Elite reunited again, Bunny was absent. Her actua whereabouts are unknown but is possible than she was free to returned to her home universe.
In the same way than the original Elite was a stand-in for the Authority, Bunny was a doppelganger for the Authority's own living ship, the Carrier.
In Superman vs. The Elite animated movie, Bunny made an appearance in a similar role as the comic where it was the largest known silicone based bacteria colony in the universe. At some point, the Elite captured it and called it Bunny where they determined it had the ability to feel which led to them jettisoning her emotion nodules. Bunny was actually able to reside within different dimensions and had the capacity to teleport people around the world. Manchester Black used her as the primary mode of transportation for himself and the Elite. During the confrontation with Superman, Bunny arrived in Metropolis where unknown to the Elite she had secretly been contacted by one of the Super-Bots who promised her the means to return to her home dimension. Thus, she helped Superman in his plan to end the threat posed by the newfound popularity of the Elite and their more vicious approach in handling threats.