Well said. Some particular examples of things we're looking for in this case, however, include supersymmetry, strong gravity / extra dimensions, and dark matter

Well said. Some particular examples of things we're looking for in this case, however, include supersymmetry, strong gravity / extra dimensions, and dark matter. All of these theories generally can be contrived one way or another to alter the predictions of the standard Higgs theory. But we are also looking everywhere we can and trying to do precise measurements because we may find something surprising as well!