The potential for the Higgs field is standard 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. But this may lead to differences in the derived particle masses.
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 ules out the cubic term.