Actually all kinds of people have tried to develop models of computation that can do more than a Turing Machine can. Just one little problem: you can't build them.
If you figure out how to build one, and you're really good with patent litigation, NDAs and building businesses generally, you'll be a rich man. If you figure out how to build one, and you're not really good with those things, someone else will be a rich man.
But either way, the trick isn't in the imagining part. The trick is in the doing part.
There is no 'proof' that there isn't a more powerful computational model, but, in agreement with the above, its the Church-Turing Thesis that claims there isn't. The Church-Turing Thesis is called a 'thesis' because its not something that can possibly be proven. But people have been trying to find an exception for a long time, and haven't managed. All of the proposals involve things like 'infinitely fast computation', and 'computing over the reals', 'oracles -- which just happen to know the answers to some key incomputable results', etc. All of which involve non-physical phenomena, aka magic (hence by the way, the term oracle).
It is modestly surprising that the Turing model of computation is so powerful, and there is a natural tendency to 'want to do better'. If you're already rich, have at it. If you have to work for a living... probably best to find an easier problem.