Umm not quite, in one of Lucete's tapes she states that she engineered the power of quantum physics into elizabeth, and near the end of the game right after you get out of the comstock house elizabeth says that she realizes that non of it is her fault because she cannot create universes, she can only enter new ones, look it up.
Just to clear things up, the twins infused the power to use tears into elizabeth by a machine they invented, she did not have the power naturally, and elizabeth cannot "change" dimensions, it clearly states that in the game. She can only enter different ones, and until she finds the infinite doors, she cannot choose what dimension to go into, she just has to use the one thats there. And sadly, I dont believe there is a connection between columbia and rapture, there just happened to ...
I see, what i was saying is I assumed that the name Annabell was referring to Booker's Anna, which would be nearly impossible. I definitely understand why she said that now, thanks.