I just read a fascinating article about the square root of -1 (or commonly known as i) over at one of the more geeky blogs I read: Good Math, Bad Math. History + Math + Complex Concepts That Make Me Feel Smart = Happy Mike:
It got its name as the imaginary number as a result of a diatribe by Rene Descartes, who believed it was a phony artifact of sloppy algebra. He did not accept that it had any meaning at all: thus it was an "imaginary" number.
Once the reality of i as a number was accepted, mathematics was changed irrevocably. Instead of the numbers described by algebraic equations being points on a line, suddenly they become points on a plane.
I'm not sure if I understand i any better, but I certainly enjoyed reading about it.