Why has the Higgs potential the form it has?

The potential for the Higgs field is a quartic one (Mexican hat). Is this done for simplicity or are there fundamental reasons for this choice? I can imagine further contributions to this potential without altering the essentials, which may lead to differences in the derived particle masses.

-
I always thought is was more "urban sombrero" shaped. –  Jiminion Mar 4 at 15:50

Cubics would also do but only in combination with quartics as a pure cubic potential is not bounded below. A further discrete symmetry assumption (e.g. symmetry under $\phi \mapsto -\phi$) rules out the cubic term.
Regarding the first sentence, you'd have to say in $d=4$. –  JamalS Mar 4 at 14:37
It means that it is the higher power that works : $\phi^{5}$ would not be renormalizable –  agemO Mar 4 at 15:55