I was wondering, how would could I make a drawing of the Higgs potential aka "mexican hat potential". I am quite new to Mathematica and don't know where to look to learn how to implement such a complex function.
It is probably a common student misconception that it is "such a complex function". In actuality, it is quite simple. You should read: A pedagogical example: the Mexican hat potential. Very roughly you get the Mexican hat from interplay between two power functions:
Now rotate with
There is a nice Demonstration "The Higgs Particle" by S. M. Blinder. You can often find relevant things at the Demonstrations Project - code can be downloaded. A good start for your homework.