How do models of hypercomputation overcome the Halting Problem?
In the paper On the power of multiplication in random access machines
 it has been proven by Hartmanis and Simon that, if we add unit cost multiplication instruction in a RAM (called MRAM) then for this model P = NP. In addition the languages decided in polynomial time in the MRAM model are exactly the languages in PSPACE. 

As stated in the paper, this results shows that multiplication have the same complexity as addition iff P = PSPACE.

A more related result I have heard of, is that if we add a division instruction with infinite precision in a RAM then we can solve undecidable problems. However I could not find the paper that proves this result. If anyone is familiar with it please comment and I will update the answer.