Drag a complex number
The blue dot is z. Drag it anywhere. The other dots are what you get by multiplying z by i, squaring it, and negating it. Try putting z on the unit circle, then outside it, then on the real axis.
What's going on
A complex number a + bi is just a point: a steps right, b steps up. Multiplying by i sends (a, b) to (−b, a), which is exactly a quarter turn anticlockwise. Do it twice and you have turned half way round, which is the same as multiplying by −1. That is what i² = −1 means in pictures.
Squaring is more dramatic. Measure z by its length and its angle from the positive real axis. For any z other than 0, z² has the length squared and the angle doubled. Zero is the exception: it has length 0 and no angle, because it points nowhere. Drag z around the unit circle and watch z² go round twice as fast while staying on the circle. Drag z inside the circle and z² shrinks toward 0.
Sheet: Imaginary numbers are real.