I believe a Turing machine is basically a model used to represent known computer capabilities (the ability to solve certain kinds of problems). Turing machines have limits in that there are problems that they simply cannot solve (or at least cannot solve in a practical time period). Hypercomputation is a new term to me. Seems that it is basically a concept for a computer that can do that things that Turing machines (and therefore current computers) cannot. One example might be to run an infinite number of computations in a finite period of time. I doubt such a thing is possible in our universe.

However, if hypercomputation proves to be possible, then all that would falsify is the hypothesis that we are being simulated by a Turing machine. It could still be possible that we are being simulated on a hypercomputer of some sort.