Why is the sqrt. of negative one relation important for imaginary numbers? 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).
 A: Because historically, the question that needed to be addressed was not "What definition can we give to a second continuum of numbers?" with the answer "Let's say they're multiples of $\sqrt{-1}$."  Rather, it was "What can we do when multiples of $\sqrt{-1}$ emerge from algebraic computations?" with the answer "Let's treat them as a second continuum of numbers."
A: The modern approach to complex numbers doesn't use square roots at all. The modern approach is to say $i^2=-1$ rather than $i=\sqrt{-1}$. It is more indirect, but actually makes a lot more sense once you get used to working with complex numbers.
How do you work with this? Essentially, complex numbers are polynomials in the variable $i$. They behave like any other polynomials we know and love. With one exception: whenever you see $i^2$, you're allowed to substitute it with $-1$ (this includes cases like $i^3=i^2\cdot i$ and $i^{1000}=(i^2)^{500}$).
That's it. That's what the complex numbers are.
Note that $i^2=-1$ is essential to make complex numbers something other than pure polynomials. So powers of $i$ must be related to real numbers. (Many other quadratic relations can be used instead if you wish, like $i^2=i-1$; basically, any relation that doesn't have a real solution if you interpret it as an equation will give you the complex numbers as a result, but they won't look exactly like the standard complex numbers.)
My gut says it's a better way to teach complex numbers as well. But I haven't tried that enough to actually say for certain.
