It's not arbitrary. It comes directly from the axiomatics.

Axiomatics are entirely arbitrary if you will. Thats the whole point: I restrict myself to some basic rules only because that allows me to show other potentially useful things, consequences and applications.

No it's not. i is a number such that i^2 = -1 by definition.

And in some contexts -1 is rotating by 180 degrees, which happens to be 290, or i*i.

Imaginary numbers are defined. Imaginary numbers are not rotations or anything else that I keep hearing. Those are all properties of the fact that we define an imaginary number to be z = a + bi, where a and b are real numbers, and i^2 = -1. That's it.

If someone asks what is an imaginary number the correct answer is "z = a+bi, where a and b are real numbers, and i^2 = -1".

You're thinking of complex numbers.