Mandelbrot Set
Drag the magenta/pink point around on the complex plane. The green points represent successive iterations of the , rule, where c is determined by the magenta/pink point.
I used this construction in a quick "Complex numbers are not as lame as you think" lesson for my PreCalculus class. In order not to hint at how the lesson would unfold, I avoided providing vocabulary such as "cardioid" and "Mandelbrot Set" within the construction.
Addition of the "trace orbit point" and "clear trace" options were inspired by this Numberphile video, and the featured Geogebra construction by Ben Sparks. With this option checked, points that cause the orbit to diverge (i.e. they spiral outward) leave a red trace, and points that do not cause the orbit to diverge (i.e. they spiral inward) leave a light green trace.