"How to Build a Superluminal Computer":

...Having created a medium in which the refractive index is less than one, Putz and Svozil's idea is simply to immerse a computer in it. That simple act (and presumably some clever design to create an optical computer in the first place) would allow superluminal computation to take place.

Assuming that this device could actually be built, what could you do with a superluminal computer? That's a good question that Putz and Svozil do not address directly. They say such a device would fall into a class of processing machine known as hypercomputers. These are hypothetical devices more powerful than Turing machines, that allow non-Turing computations. They were first discussed by Alan Turing in the 1930s.

In theory, hypercomputers can compute certain kinds of otherwise noncomputable functions. That sounds handy but even though there are uncountably many non-computable functions, it's actually quite hard to come up with an example of one that might seem useful. If you have any ideas, post them in the comments section.