Complex numbers have a terrible name. They're not complicated but the name puts people off.
They're actually simple, the name is an accident of history and if anything refers to the fact that they are made of up 2 independent parts. They would have been better called composite numbers but that name is taken!
This blog entry is a gentle and interesting introduction to complex numbers
I like it because it introduces ideas by relating them to things we are all familiar and comfortable with. And like the best tutorials it makes excellent use of visual explanations.
In particular I like his gentle introduction to arithmetic as geometric transformations. Multiply by 3 means magnify by 3. Multiply by -1 means flip across the origin point on the number line. Multiply by -3 means do both scaling and flipping.
It's natural then to ask which transformation turns 1 into -1 when applied twice. A 90 degree rotation counterclockwise in a two dimensional plane works, and is in fact a great way to think of the complex i operator. Suddenly i2 = -1 makes so much more sense!
What a great introduction to the otherwise strange i, and a great encouragement to think about apparently innocent looking mathematics as geometrical transformations.