A polynomial of degree two
Of course the previously shown square function is also a polynomial of degree two, but now we shall examine a bit more general one.
\(f(z) = z^2-1,\quad z \in \mathbb{C}(0,2)\)
The roots are at 1 and -1. One could discover these where the four colors meet on the picture. These zeros are both of multiplicity one. (Let us go round them staying sufficiently close and observe the colors.)
Changing specific parameters successively and plotting each of the resulting functions one after another, we could create nice animations. One related example can be seen below, and can be downloaded by clickink on this link.
In this video we move the two roots (of opposite values) of a polynomial of degree two along the unit circle, i.e.: \[ f_{\varphi}(z) := (z-t_0)(z+t_0),\quad t_0 = \exp\, i\varphi,\quad \varphi \in [0,\pi]. \]
Zeros in the animation.