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But could there potentially be a medium that increases the speed of light? - Maybe it is the defining property of "Dark Matter" :)

Nope. It has been shown that dark matter doesn't interact with the electromagnetic spectrum. I do not believe the is any current theory that allows c as a fundamental limit to be broken.

I note that in this whole discussion thread that "super-luminal" velocity as we understand it has been avoided, I suspect that's mainly because even a partial explanation would make your brain hurt (it did mine!) Although not quite the same but far more approachable, Cherenkov radiation (seen as the funky blue glow in nuclear storage ponds) can be considered the result of superluminal particles ... which *do not* travel faster than light but travel faster than light *in that medium*. c, the speed of light in a vacuum is a constant and an ultimate limit in any frame of reference. The speed of light at any point in spacetime is variable and only has an upper bound of c.