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.
In four dimensions, the quartic is the worst interaction between scalar fields that is renormalizable (by power counting arguments).
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.