It's also perfectly possible to draw maps of places that don't exist in reality, i.e., that only exist in somebody's brain.

A good point. So the real numbers exist in the same sense that anything else they might describe does.

My original point was really a comment on the way that the subject is often taught: the real number system is not introduced as some arbitrary set of rules that is as good as any other (though of course it is); it is introduced as if it were as incontrovertible an idea as, say… integer addition. I don’t think it is. It can perfectly well be argued that it makes sense and is useful, but its naturality is another matter. I personally find the leap from R to C less controversial than that from Q to R.

Yes, I agree. Real numbers are a quite nebulous construct for which the need is a bit hard to see. Complex numbers are basically a hack to solve quadratic equations in all cases.