>beyond a certain level of complexity you'll somehow automatically get consciousness

Right, that's the argument, but I don't believe this at all. Consciousness is a computation therefore all computations are conscious. That doesn't follow. In any other philosophical or scientific discourse that sort of naive conceptual inversion would be laughed out of the room. Horses are animals therefore animals are all horses. It's absurd.

Personally I think consciousness arises out of reflection on an internal model of one's own thought processes and behaviour. See Douglas Hofstadter's "I am a strange loop". I see no reason why a system like that wouldn't be tractable to computation.