Out of all the people Bellatrix hates, Alice is the one she hates the most.
Alice is the epitome of perfection, well-mannered and the ever obedient Pureblood princess. She makes Bellatrix feel like she's not even good enough to coexist next to her. She thinks Bellatrix ought to stop 'messing around', find herself a husband and forfill her duty as a Pureblood wife. She gets praised a lot, but Bellatrix doesn't want the kind of praise Alice is getting.
Bellatrix wants to get praised for her skills, not her looks. She's ambitious and a rebel and she wants to prove to society that just because she's born a girl, that doesn't make her less of a force to be reckoned with. She's been in Alice's world, but it's a world she doesn't belong in.
Alice threatens to leave if Bellatrix indulges in her fascination for Lord Voldemort.
Bellatrix takes the Dark Mark partly to defy her, but contrary to her threat, Alice leaves only temporarily.
When the first Crucio hits its target, Bellatrix can't figure out if it's Alice who's screaming, or if she is. But she soon discovers that Alice is weaker than her, that she can't stand the pain as Bellatrix can, and that torturing Alice will bring Bellatrix peace.
Bellatrix skills her torturing to perfection, and every time Alice writhes and cringes and cries ear-splitting cries of agony that are sweet music to Bellatrix' ears.
The Dark Lord praises her - for her skills, not for her looks - and she dedicates herself even more vehemently to his cause, proud to be his most trusted lieutenant.
For a while, Alice disappears completely out of her life. Bellatrix never stops inflicting pain though, because Alice hates that and she hates Alice and when she stops, Alice might come back to criticise and ridicule her again. As long as she's inflicting pain, she can keep Alice at bay.
"Alice, take my wand!"
Bellatrix stops dead in her tracks and watches Longbottom throw his wand towards Alice. Not the Alice she's grown to hate, but another Alice. The first blow nearly sends her flying and Bellatrix' eyes spit fire. She did not defy one Alice to be killed by another.
She holds her Crucio longer than usual, everything to wipe that confidence of Alice's - the other one - face. Alice shrieks, then cries, then screams, then sobs, then screams again.
Bellatrix lifts her Crucio for a second, only to catch her breath, and in that second Alice - the one she hates - reappears, drumming her fists on Bellatrix' chest, kicking her with a vengeance and commanding her to stop.
But Alice doesn't understand. Alice doesn't have anything to command anymore. Bellatrix refuses to let her take control again.
She responds to Alice's assault by the only way that has proved effective in the past: by inflicting more and more pain. It's the other Alice, but the effect is the same.
By the time the Aurors come for her, Bellatrix is laughing maniacally. "You'll never win Alice, your days are over! You'll never again be stronger than me!"
Alice - both Alices - remains silent.
Bellatrix is sent to Azkaban. In passing a looking glass, she catches a glimp of Alice, who's mocking her - You'll never get rid of me - and she's falling, falling, falling, deeper into the abyss until she screams.
Everyone always said Bellatrix could withstand pain like no other. Some even say that she revels in it and that the way she inflicts it on herself is a sign of her deranged mind.
They don't understand though, that it's the only way to make Alice shut up.
A/N: I've never written schizophrenic!Bellatrix before, but I'm actually quite pleased with how it turned out. :)