Answer by Tikhon Jelvis:
The best intuition I've seen is that complex numbers are about rotation. Basically, they take the 180° flipping that negation for real numbers represents and generalize it to rotation in a circle.
Perhaps a more general (but more vague) description is that complex numbers generalize 1-dimensional real numbers to 2 dimensions.
Take a look at, a beautiful aricle that illustrates this idea and shows how it relates to fractals. [The visualizations require a browser and graphcs card with good support for WebGL.]