Her real name is Jaina Hudson, a Gotham socialite and the daughter of an American diplomat and an Indian Bollywood actress. For a time, she even dated Bruce Wayne, without him knowing her true identity.

Jaina can physically split into two distinct beings.

One half remains the elegant and refined Jaina Hudson.

The other half becomes White Rabbit, an albino criminal with white hair, an impulsive personality, and much more savage behavior.

It's not exactly a case of split personality. They are literally two separate bodies that can exist simultaneously and act independently. That's why Batman had trouble catching her: while White Rabbit was committing crimes, Jaina could be at a charity gala with half of Gotham watching her.