Somewhere I read something like "a hypercomputer rotating around a rotating black hole" would have some esoteric properties e.g. would produce other answers than other hypercomputers and other computing. Is there any support for this, do I remember incorrectly, or is this claim known to be nonsense?
There are actually sort-of-researchy things one can say about rotating black holes. Namely it is known that sufficiently quick rotation produces time-like simple closed curves, leading to interesting questions about how to model time-travel paradoxes in complexity theory; see Aaronson and Watrous, "Closed Timelike Curves Make Quantum and Classical Computing Equivalent", arXiv:0808.2669
I have no particular opinion about combining hypercomputers with time travel that does not generalize to my opinion about hypercomputers in general, so probably it's best if I just point you to the one paper I could find that both cites Aaronson and Watrous and discusses hypercomputers:
Hajnal Andréka, István Németi, and Gergely Székely, "Closed Timelike Curves in Relativistic Computation", Parallel Processing Letters, 2012 and arXiv:1105.0047