The inflection points of real continuous functions have a relatively clear visual interpretation - they are the points when the function's graph moves from convex to concave, or vice-versa. Conversely, the critical points of a holomorphic function can also be visualised informatively. They're the ramification points, or the points around which the winding number is non-zero, and so on a domain-colouring plot of the function they are easily identified as the points where the plot moves locally through the full spectrum, or as the convergence points of the lines of equal phase. However, I have not come across a visual intuition for the inflection points of holomorphic functions, and would be interested in any suggestions.
This came up in teaching high school mathematics to a few advanced students, where I wanted to emphasise the importance of the inflection point in the study of cubic functions. Obviously, the original reason complex numbers attracted popular interest was the usefulness of passing to a complex domain to explain the properties, and find the roots, of real-valued cubic polynomials, so I was interested in what the inflection point of a complex cubic would 'look like'
Here are a few quick plots in Mathematica.
Later, may update with more details.