We have no idea whether that is 'more complicated than binary code' or not. If you have enough binary then you can have endless complexity, I would suspect. It looks far more complicated than computers, but that is not the same thing.

All that computers do is to take information and generate a response. That's more or less what we do. It may be that 'free will,' our capacity for seemingly infinitely varied responses to situations, is simply a product of the infinitely varied situations that we are able to distinguish.

It is also perfectly reasonable to suppose that odd memories that strike us from time to time for no apparent reason are merely a consequence of an imperfect brain - it has retrieved something useless by mistake because it is slightly wrongly wired up, and it really meant to retrieve something else that would have been natural and obvious to you.

As you say, this is all 100% hypothetical, but I think that if there are credible solutions that do not require the existence of a mind that is separate from the brain then we shold be very reluctant to imagine that a separate mind truly exists, because it does not tie in with what we can see. It apparently ignores the laws of physics (by definition, being metaphysical), which seem to apply in every other known situation, and we'd need a very good reason to suppose that we should do that.