You wrap this in a P/NP question but you're really asking whether the Church-Turing thesis is true.
Btw. I completely agree that complexity theory should be "rewritten", or rather written down in a mathematical way in the first place. For some reasons that's a field which got so seperated from language theory that the only ever write mostly natural language proofs. I'm not saying you have to do formal logic proofs of your stuff, but one textbook which does this once would help.
Also, I don't know what you use the phrase "computational power" here, when that reeks of just energy (while you try to break from the Turing complete framework) and also I don't think, again, rewriting the existing theory is the best start to get at novel models.
Actually, maybe the physics of time can lead to hypercomputation frameworks that help you there.
https://en.wikipedia.org/wiki/Hypercomputation 
