I think whether you call it a thought process or not comes down to definition of what you mean by that. It's definitely intelligence, and there definitely is a process.
So I wouldn't have a problem calling it a thought process. But it's not self consciousness yet. But we may not be very far from it.
It's amazing the progress that has been achieved the past decade.
When I predicted 2035 as a point where we could possibly achieve strong AI, it was at a point where we'd had 2-3 decades of very little progress. But I've always been certain that the human brain is a 100% natural phenomenon, and the function of it can be copied, just like with everything else in nature. And when that is achieved, there will still be room for improvement.
As a natural process, our brain is built on the physical properties of atoms, so IMO it's only a matter of time before we have an artificial intelligence that is just as valid to call self conscious as ourselves.