TITLE: Why is the complex plane shaped like it is?
QUESTION [71 upvotes]: It's always taken for granted that the real number line is perpendicular to multiples of $i$, but why is that? Why isn't $i$ just at some non-90 degree angle to the real number line? Could someone please explain the logic or rationale behind this? It seems self-apparent to me, but I cannot actually see why it is.
Furthermore, why is the real number line even straight? Why does it not bend or curve? I suppose arbitrarily it might be strange to bend it, but why couldn't it bend at 0? Is there a proof showing why?
Of course, these things seem natural to me and make sense, but why does the complex plane have its shape? Is there a detailed proof showing precisely why, or is it just an arbitrary choice some person made many years ago that we choose to accept because it makes sense to us?

REPLY [15 votes]: Riemann

made it into a sphere: