This is a fantastic explanation. I've always had trouble "getting" imaginary numbers.... even though I've had to use the fairly often as an Electrical Engineer. This is the first time they've made intuitive sense to me.
EDIT: I guess I learned it in high school though, and the math teacher probably didn't know this either.
Point being, everyone should come across this at some point. It would be beneficial to many math students.
What blew my mind at the time was the exponential notation for the unit phasor: e^(i * x). It turns out that e^(i * x) = cos(x) + i * sin(x) because that's just the way the math works out, and it's trivial to work it out yourself by looking at the Taylor series expansions of the three terms.