Is there any geometric interpretation or significance of the complex roots of a derivative?

Question

I was doing some reading online when I stumbled here and learned about this geometric way of viewing the complex roots of a function. It got me thinking; the zeros of the derivative of a function $f$ indicate points at which the slope of $f$ is $0$. Is there some kind of geometric interpretation of the complex roots of a function's derivative?

Answer 1

The graphical interpretation of complex roots, while cute, only applies to second order polynomials with real coefficients. And if you have a second order polynomial with real coefficients, its derivative will have no complex roots. SO your question won't elicit any equally cute interpretation.

Answer 2

I don't know if this is the kind of answer you're looking for, but if $f\colon\mathbf{C}\to\mathbf{C}$ is a complex analytic function, then the zeros of $f'$ are the points where the mapping $f$ is not conformal.