First I'll state a statement that I hope is false, but I do not know if it is: "Complex analysis is used a lot compared to analysis over other fields (as in it gives a lot of results like the prime number theorem and such that other fields don't)."
If my statement isn't true, can you provide examples of extraordinary results of analysis over other fields?
If it is, why is it that complex numbers give those results and other fields don't?
My guess is that it's the only algebric closure of R, but why is that important for deriving theorems in number theory?