Why is the square root of negative one an important relation for imaginary numbers and quaternions, etc.?
I mean, if we are going to have imaginary numbers, couldn't they be imaginary without needing to be defined in terms of the square root of negative one?
I.e. why couldn't we have imaginary numbers without them having any definition in terms of a relation to the real numbers?
Seems to me that you could say imaginary numbers are based on the square root of x, where x is some number that's not on the real number line (but not necessarily square root of negative one—maybe instead, 1/0).