To be clear: this isn't a question about whether or not choosing one necklace or the other has an impact on gameplay, or a duplicate of Elizabeth's Brooch - follow-up question, since (please read in full)...
Elizabeth's Brooch - follow-up question refers to the scene where Booker enters the lighthouse and is drowned by the other Elizabeths (all without a necklace), implying that "his" Elizabeth hasn't entered with him at all (possibly to avoid the threat to her very existence should she enter since she would be killing her father before her own conception). The below refers to the scene in the rowboat before they enter the lighthouse where Elizabeth is seen with a different necklace (not the lack thereof)--this is therefore a different question with different implications (i.e. that the Elizabeth guiding Booker through the Sea of Doors to the drowning already is a different Elizabeth for some unknown reason).
I personally chose the bird for Elizabeth during a second play-through, and tried to pay attention to it the whole way through, now aware of its significance.
I noticed the following:
when I was rowing to the lighthouse with her and the Luteces for the third and last time before entering the lighthouse at the very end of the game, her pendant was briefly changed to the cage right up until Booker climbs the ladder of the dock (i.e. right after you cross the tear in the wall of your apartment of "20 years later" to join the Luteces waiting for you, they comment on the mark on your hand, and you board the rowboat for a third time).
The game gives you plenty of time to catch the switch during this scene:
as light reflects off the pendant multiple times while you're rowing to the dock and Elizabeth says "Booker wake up, this is where it started" to which he replies "I sold you... I sold you...", you can clearly see the pendant is now a cage when it should be a bird (in my case).
My question is: does this mean the Elizabeth that is with you during that very short segment is actually another Elizabeth? And, if so, what possible implications does it have?