Human brains and experiences seem to be constrained by the laws of quantum physics, which can be simulated to arbitrary fidelity on a computer. Nit sure where Godel’s incompleteness theory would even come in here…
I think there is abundant evidence that the answer is ‘no’. The main reason is that consciousness doesn’t give you new physics, it follows the same rules and restrictions. It seems to be “part of” the standard natural universe, not something distinct.