This should probably be in the comments but...
I'm going to say "Baptism".
From reading the Bioshock Wikia Article about Booker, it seems that Booker is repeatedly given the option to be Baptized into the Christian religion and, in his version of reality, opts out on various occasions and, apparently almost does get Baptized but is knocked out cold.
Towards the end of the Wikia article, in the last paragraph before the Epilogue section it reads:
Booker then realizes that the only way to erase the atrocities committed by Comstock (himself), and the harm visited upon Elizabeth, is for him to die before he can accept or reject his baptism. Thus cancelling out either outcome.
This sounds to me like a two sided coin... one that he can't pick up because doing so creates two flawed people who can't exist without destroying Elizabeth, his daughter.