Alicia, played by Mick Jagger's ex Jerry Hall, is the girlfriend of Carl Grissom and secret lover of Jack Napier. They both call her "Sugar Bumps," and it's strange that two men would have an identical pet name for a woman. We thought her name was Sugar Bumps until Joker says her name is Alicia late in the movie… so late she's already dead by the time he tells us.
We're not sure what she sees in Joker, other than an opportunity to be close to power and money. He's rude to her as Jack Napier, scars her face as Joker, makes her wear a mask, and drives her crazy until she commits suicide. Then, he callously tells Vicki:
JOKER: Alicia threw herself out of the window[…] But you can't make an omelet without breaking some eggs.
With Sugar Bumps a bump on the pavement, Joker wants to replace her with Vicki – another blonde bombshell. To Joker, these women are interchangeable and disposable. He'd probably call Vicki "Sugar Bumps," too, but then she'd kick him in his sugar bumps.