Water from bathroom faucet smells like rotten eggs
A rotten egg smell in water is usually from high sulfer content in the water.  Not unheard of in very hard water. Where it could possibly be coming from in the pipe perplexes me though. The fact that it only comes out one faucet is odd.  If both the hot and cold smell bad at only that faucet and no other than the problem must be in the faucet itself, since if it was in the water heater all the hot water in the house would smell and the hot and cold pipes are 2 different pipes until they get passed the supply lines below the sink.  

I would shut off the water below the sink, open the valves to drain out any water that might be sitting in the pipe and then see if the faucet outlet itself smells odd.  If there is a screen at the mouth of the faucet remove it and see if there is debris inside.

This is hardly an answer but I'd target the faucet or something trapped inside it as the problem.

Love to hear what the problem ends up being.