> So what you are saying is that consciousness is a property that emerges from a complex neural network that our brain is

Sort of, minus the word "is" in "consciousness is..." since nobody, as you point out, can say what consciousness "is." Without some qualifier. "I hypothesize that..."

> But that would also mean that consciousness cannot invoke any quantum mechanics

Well, consciousness isn't known to be physical, yet does influence the physical world (probably), which is a major criticism of dualism from a materialist perspective - clearly, if it's influencing the material world, it's material! But, we don't know, so, no, it doesn't necessarily mean that consciousness can't invoke quantum mechanics if it's not physical.

> Claude effectively claims to be conscious, why should we not believe him?

I agree generally that we don't know enough about consciousness to "prove" it exists, but I disagree insomuch as I don't like to engage in cleanroom absolute theoretics about it - e.g., yeah I can't prove what consciousness is, but I accept that you're probably conscious, and that my friends I know in real life absolutely are, and Claude isn't because it's basically a Chinese Room. I don't have rigid theorics around this but I don't really believe anyone does, so I guess I'm just vibing about it for now.