If you really want to "demystify" complex numbers, I'd suggest teaching what complex multiplication looks like with the following picture, as opposed to a matrix representation:

If you want to visualize the product "z w", start with '0' and 'w' in the complex plane, then make a new complex plane where '0' sits above '0' and '1' sits above 'w'. If you look for 'z' up above, you see that 'z' sits above something you name 'z w'. You could teach this picture for just the real numbers or integers first -- the idea of using the rest of the points of the plane to do the same thing is a natural extension.

You can use this picture to visually "demystify" a lot of things:

• Why is a negative times a negative a positive? --- I know some people who lost hope in understanding math as soon as they were told this fact
• i^2 = -1
• (zw)t = z(wt) --- I think this is a better explanation than a matrix representation as to why the product is associative
• |zw| = |z| |w|
• (z + w)v = zv + wv
• The Pythagorean Theorem: draw (1-it)(1+it) = 1 + t^2 etc.

One thing that's not so easy to see this way is the commutativity (for good reasons).

After everyone has a grasp on how complex multiplication looks, you can get into the differential equation: $\frac{dz}{dt} = i z , z(0) = 1$ which Qiaochu noted travels counterclockwise in a unit circle at unit speed. You can use it to give a good definition for sine and cosine -- in particular, you get to define $\pi$ as the smallest positive solution to $e^{i \pi} = -1$. It's then physically obvious (as long as you understand the multiplication) that $e^{i(x+y)} = e^{ix} e^{iy}$, and your students get to actually understand all those hard/impossible to remember facts about trig functions (like angle addition and derivatives) that they were forced to memorize earlier in their lives. It may also be fun to discuss how the picture for $(1 + \frac{z}{n})^n$ turns into a picture of that differential equation in the "compound interest" limit as $n \to \infty$; doing so provides a bridge to power series, and gives an opportunity to understand the basic properties of the real exponential function more intuitively as well.

But this stuff is less demystifying complex numbers and more... demystifying other stuff using complex numbers.

Here's a link to some Feynman lectures on Quantum Electrodynamics (somehow prepared for a general audience) if you really need some flat out real-world complex numbers

http://video.google.com.au/videosearch?q=feynman+auckland&filter=0&start=0#