I think we're both on the same page, maybe I'm just saying it wrong. Yes, I believe BaS' "Booker" was an alternate timeline version.
Earlier I was saying that I thought I read somewhere that this was a "new" Elizabeth, it was implied that this was not "Infinite" Elizabeth. But, regardless, I believe that to be false. BaS's Elizabeth is Infinite Elizabeth. I'd imagine Elizabeth is a constant.
Ok, cool - yeah, lot of possibilities with her character