The problem with this way of thinking is that "the actual mathematics of our universe that I care about" doesn't (yet?) exist. The definitions you find badly chosen have been thought about, hard, by many generations of very smart people. Some with the same basic reaction to all this you have. And yet, after all that, the definitions remain. And the alternatives proposed, at best are not compelling.

So why? Real & Complex analysis are by far the best way humans have yet come up with to model certain things, particularly modeling a large subset of physical interactions in the universe.

So to some degree we are stuck with this. Mostly it's the fact that fields like R an C are actually pretty strange objects. Strange in ways that are hard to get your head around, and that leak out into any mathematics that you build on top of it.

So what to do? There is beauty there as well. And although we have tricks to hide most of this, most of the time, it's not a bad thing really that when you look closely the weirdness is apparent.