Before a computer is assembled, it's a pile of useless wires and hardware. Put it all together and the whole is much greater than its parts, in that it can do things like beat the best chess player in the world. Conversely with the human brain, severe enough head injuries can cause profound changes in personality. Doesn't this "whole much greater than the sum of parts" not prove that dualism fails Occam's razor? I mean, if there was a soul independent of brain matter, where does it go after severe head injuries? By all accounts, people are not who they used to be after such unfortunate losses.
You are right that it is no argument for dualism that brains can do things that single neurons can't. The most powerful case for dualism is probably the enormous difficulty in seeing how facts about conscious experience could be purely physical facts, however complex. There is no fundamental conceptual difficulty in seeing how bits of silicon and wire could be put together in a way that yields a computer that can beat humans at chess; but it's much harder to see how those bits could be put together to generate conscious experience. Even that, however, is not the big problem, because many dualists would hold that mental states are caused by physical states. The big problem is seeing how the conscious state could itself be (not just be caused by) something entirely physical. This is the basis for the controversial and stimulating Knowledge Argument for dualism, due to Frank Jackson. No matter how much a deaf person knows about the physical properties of the sensation of sound, indeed even if he...