Jesus Christ, the complex plane. I half-remember that.

Eliezer, this may come as a shock, but I suspect there exists at least some minority of individuals beyond just me who will find the consistent use of complex numbers at all to be the the most migraine-inducing part of this. You might also find that even those of us who supposedly know how to do some computation on the complex plane are likely to have little to no intuitive grasp of complex numbers. Emphasis on computation over understanding in mathematics teaching, while pervasive, does not tend to serve students well. I could be the only one for whom this applies, but I wouldn't bet heavily on it.

Thankfully, there's already a couple of "intuitive explanations" of complex numbers on the 'net. And I dug up the links. And the same site has a few articles about generalizations of the Pythagorean Theorem, on a related note. Basically, anyone who is having any trouble with the mathematical side of this is likely to find it a bit of a help. It's also a lot like overcoming bias in its explanatory approach, so there's that.

Imaginary/complex numbers:

http://betterexplained.com/articles/a-visual-intuitive-guide-to-imaginary-numbers/

http://betterexplained.com/articles/intuitive-arithmetic-with-complex-numbers/

General math:

http://betterexplained.com/articles/category/math/

I should probably also take this opportunity to swear up and down that I'm not trying to generate ad revenue for that site, but you'd have to take my word on it. I might also add that I'm quite certain I still don't understand complex numbers meaningfully, but that's a separate thing.