After the death of Comstock, Elizabeth starts jumping dimensions with you and show you how there are infinite amount of dimension, created from single decisions. Although Comstock is dead in your world, he's still alive countless amounts of times. So Elisabeth takes you a defining point in your life. After the battle of Wounded knee Booker has the decision to become a born-again Christian, or run away. If he runs, he remains Booker Dewitt. If he stays he becomes Comstock. Elizabeth takes you here because it is the single consistent variable in all of the infinite worlds. If Booker dies here, Comstock will die. Colombia won't be created and hence New york will be saved.
The 'give us the girl and wipe away the dept' thing is Comstock using Lutece interdenominational machine to take Booker's child away (Who turns out to be Elizabeth). Booker makes the story of being in dept to gangsters or something up in his head because the Lutece twins (who aren't really twins) take him from his world and into there's. Since he didn't have a past there, he filled in the blanks himself.
As for Booker still being alive with his daughter.
The Lutece's did it.
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