Python 2.7 – 2934*
Oh how much this first one sucks. Pretty ugly, this one is. 63 chars.
''print''.join([bin(~0)[3:] if x == '0' else bin(~1)[4:] for x in '']) This one is a bit better but still not that fancy. 44 chars.
''print''.join([str(int(not(int(x)))) for x in '']) Since int(x) and 1 returns int(x) if it's not 0 and otherwise False. The solution can be further reduced to 36 chars.
''print''.join([str(1-int(x)) for x in '']) Since join() takes a generator the brackets can be removed. 32 chars.
''print''.join(str(1-int(x))for x in'') And backticks can be used instead of str()
''print''.join(`1-int(x)`for x in'') Reduced to 44 from 2934 thanks to pointers from @TheRare
Finding one's complement is difficult in python since bin(-int) returns -0bxxx hence the above.