Why is the complex plane shaped like it is? 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?

The complex plane is a model (or more formally, an isomorphism) with very nice properties, such as $\arg(zw)=\arg(z)+\arg(w)$ or, in more advanced complex analysis, winding numbers.
Could there be a "better" representation? Maybe. And, in any case, you should be clear about what means "better" for you, that is, what you want to do with this representation.
What is sure is that, since Gauss began to use the complex plane, deep, useful and beautiful applications have been found.
