Complex function domain colouring

I'm a tutor supporting a student as he works through his Extended Project Qualification and am looking for anyone who could help out with producing a complex colour domain plot of the function F(z)=z^(1/z), have used a couple of web apps but resolution is poor so any help would be gratefully received!

I'm a tutor supporting a student as he works through his Extended Project Qualification and am looking for anyone who could help out with producing a complex colour domain plot of the function F(z)=z^(1/z), have used a couple of web apps but resolution is poor so any help would be gratefully received!

I'm a tutor supporting a student as he works through his Extended Project Qualification and am looking for anyone who could help out with producing a complex colour domain plot of the function F(z)=z^(1/z), have used a couple of web apps but resolution is poor so any help would be gratefully received!

I'm a tutor supporting a student as he works through his Extended Project Qualification and am looking for anyone who could help out with producing a complex colour domain plot of the function F(z)=z^(1/z), have used a couple of web apps but resolution is poor so any help would be gratefully received!

Do you mean this sort of plot? I must admit, I only ever try to do them in Mathematica or Matlab. However, there is a Python plugin for Gimp (which I have not used) which looks pretty impressive, if you don't have access to these.

May I suggest that you steer your student away from $z^{1/z}$ ? Raising complex numbers to complex powers and getting a function out of it is notoriously tricky!