# How to plot on complex plane with Mandelbrot set

I essentially want to treat the Mandelbrot set plot as a normal plot so that I can plot arrows and points on top of it.

This is what I want to do but it gives me an error:

Show[MandelbrotSetPlot[{-2.5 - 2.5 I, 2.5 + 2.5 I}], Point[0 + 0 I]]


This does not work either:

Show[MandelbrotSetPlot[{-2.5 - 2.5 I, 2.5 + 2.5 I}], Point[0, 0]]


How can I display and annotate this plot?

MandelbrotSetPlot does produce a normal plot. "Normal" here means that the result has Head Graphics. Recommended reading on the term "head": Everything is an expression.

Show can be used to combine expressions which have the head Graphics.

The line below has two problems:

Show[MandelbrotSetPlot[{-2.5 - 2.5 I, 2.5 + 2.5 I}], Point[0, 0]]

1. Point[0,0] is not a Graphics expression. It should be enclosed in Graphics before using it in Show.

2. Point[0,0] is not correct syntax. Use Point[{0,0}], as shown in the documentation.

This will work:

Show[MandelbrotSetPlot[{-2.5 - 2.5 I, 2.5 + 2.5 I}], Graphics[{Point[{1, 0}]}]]


An alternative solution is Epilog:

MandelbrotSetPlot[{-2.5 - 2.5 I, 2.5 + 2.5 I}, Epilog -> {Red, PointSize[Large], Point[{0, 0}]}]


Here I also changed the colour and size of the point to make is visible over a black background.