Along with a bunch of important people who actually know how to give interviews, I answer that question in this video:
By the way, when I talk about a “minimal Higgs, that only does the part we know that something like the Higgs has to do,” I’m referring to the so-called fermiphobic Higgs. It plays the usual role of the Standard Model Higgs boson in breaking electroweak symmetry, but doesn’t couple to quarks and leptons (i.e. fermions). We already know from the way the weak and electromagnetic forces work that the relationship between them has its origins in something like the Higgs — but we have less reason to be certain that the same particle takes care of quark and lepton masses too. This version of the Higgs boson is more difficult to find, but perfectly sensible, and we’ll probably hear a lot more about it in coming years if we don’t have a big discovery this year or next.