Comment by the__alchemist

> I believe real numbers to be completely natural, but far greater mathematicians than I found them objectionable only a hundred years ago

I suspect, as you may as while, that this quote is at the core of the matter. Identifying what you find the difference between real and complex numbers are. You are inclined to split them into separate categories. I suspect you must identify the platonic (Or HTW, if that is your metaphor) property of the real numbers which the complex lack.