DON'T look at this until after you've beaten BioShock Infinite. Seriously, come back after you have.
For the rest of you that have cleared the game, you're about to have your minds blown.
I beat BioShock Infinite yesterday and after picking my jaw off the floor, I did what presumably the rest of you did and went to the Internet in order to process everything. NeoGAF has answers to a lot of your questions, and even digs into something I can't believe anyone was able to pickup on short of reading Ken Levine's mind.
Booker, Elizabeth, and the Songbird will all end up back in Rapture towards the end of the game. Booker and Elizabeth are safely inside one of the buildings of Rapture, while the Songbird is left outside underwater. The Songbird wasn't built to take pressure like one would face deep under the ocean, and you'll witness the creature get crushed all while screaming out in agony.
So okay, cool, you're back in Rapture. Nice little nod, and a great surprise. Now, what if I told you that you actually heard the Songbird screaming while playing the original BioShock?
At around the 0:52 second mark you can hear the Songbird's cries. When Elizabeth transport the group, they end up at the point in the original BioShock when players were stuck taking pictures for Sander Cohen, and it's right before Fitzpatrick dies that you hear the Songbird.
You can hear the Songbird scream more clearly at the 0:14 mark in the IGN video, and it almost matches up with the Songbird death cries in Infinite.
Sure, it's not exactly the same in the game. I imagine Irrational wanted to shorten the screams of pain and make the death quicker once they actually got around to making the scene. In any case, it all lines up and it's one hell of a nod.