I'm going to put this up as a news post, because there are still some people who seem to be confused over this. Contrary to IGN's review, you do not have to complete the game to unlock 1999 Mode in Bioshock Infinite.
As suggested in a plethora of interviews back at the start of last year, the unlock (on console at least) is a variation of the old-school 'Konami Code'.
From the main menu, all you need to do is press ↑↑↓↓← → ← → B A.
1999 Mode is a game type that harkens back to trickier times, nerfing the frequent autosaves, making ammunition scarcer, upgrades permanent, and enemies far hardier.
It transforms Bioshock Infinite into a tortuous beauty - like being kicked in the nads whilst eating exquisitely delicious cheese.