I used to feel the same way. I now consider complex numbers just as real as any other number.
The key to seeing the light is not to try convincing yourself that complex number are "real", but to truly understand how ALL numbers are abstractions. This has indeed been a perspective that has broadened my understanding of math as a whole.
Reflect on the fact that negative numbers, fractions, even zero, were once controversial and non-intuitive, the same as complex are to some now.
Even the "natural" numbers are only abstractions: they allow us to categorize by quantity. No one ever saw "two", for example.
Another thing to think about is the very nature of mathematical existence. In a certain perspective, no objects cannot exist in math. If you can think if an object with certain rules constraining it, voila, it exists, independent of whether a certain rule system prohibit its. All that matters is that we adhere to the rule system we have imagined into being. It does not exist in a certain mathematical axiomatic system, but then again axioms are by their very nature chosen.
Now in that vein here is a deep thought: I think free will exists just because we can imagine a math object into being that is neither caused nor random. No need to know how it exists, the important thing is, assuming it exists, what are its properties?
Correct. And this is the key distinction between the mathematical approach and the everyday / business / SE approach that dominates on hacker news.
Numbers are not "real", they just happen to be isomorphic to all things that are infinite in nature. That falls out from the isomorphism between countable sets and the natural numbers.
You'll often hear novices referencing the 'reals' as being "real" numbers and what we measure with and such. And yet we categorically do not ever measure or observe the reals at all. Such thing is honestly silly. Where on earth is pi on my ruler? It would be impossible to pinpoint... This is a result of the isomorphism of the real numbers to cauchy sequences of rational numbers and the definition of supremum and infinum. How on earth can any person possibly identify a physical least upper bound of an infinite set? The only things we measure with are rational numbers.
People use terms sloppily and get themselves confused. These structures are fundamental because they encode something to do with relationships between things
The natural numbers encode things which always have something right after them. All things that satisfy this property are isomorphic to the natural numbers.
Similarly complex numbers relate by rotation and things satisfying particular rotational symmetries will behave the same way as the complex numbers. Thus we use C to describe them.
As a Zen Koan:
A novice asks "are the complex numbers real?"
The master turns right and walks away.