I think you should read
https://www.physicsforums.com/insights/things-can-go-wrong-complex-numbers/
##i## is mathematically simply one of the two solutions of the real equation ##x^2+1=0##.
Or a bit more sophisticated: one representative of the two equivalence classes of ##\mathbb{R}[x]/(x^2+1)##.
That's it.
Now you might object, that this doesn't help a lot. But in the end, it is that easy. People faced similar problems (ancient Greeks), when the had to imagine ##\sqrt{2}##. It is no number. Not even finite in its decimal expression. It wasn't easy for them, as numbers always were either something to count with or a relation between two numbers, a quotient. But ##\sqrt{2}## is neither. However, they could draw it, namely as the diagonal of a square. It's not as easy with complex numbers, but in the end we found a way: the plane of complex numbers with axis ##\{1\}## and ##\{i\}##. So things are a bit more complicated with complex numbers as they are with irrationals like ##\sqrt{2}##, but therefore we can do an awful lot more with them. And as ##\sqrt{2}## solves ##x^2 -2 = 0##, ##i## solves ##x^2+1=0##. Finally it comes down to this point.
If you like, you can go back even further in time. The Babylonians needed negative numbers - nothing you can find in nature. But they did their accounting and book keeping with them. They were used for economic reasons and people didn't bother whether they are natural or not. They simply helped them doing their calculations. I'm almost sure, they did not write ##-3## but used two sides of a balance sheet, one for positive and one for negative numbers. However, this is a matter of syntax only.
And my personal most favorite achievement by mankind dates back even longer. The Indians started to use ##0## for the first time. (IIRC some seven thousand years ago.) It was by no means a trivial discovery. Someone decided to count nothing! They must have taken him as crazy. To count something that isn't there. It couldn't even be drawn. There is nothing to draw. Nevertheless, it turned out to have some practical benefits.
It was a long way to go from old India to Gauß. And in all cases, it was the benefits in calculations, that founded their popularity, not their property of being imaginable.