This question is surely somehow a product of not understanding the difference between a field in the sense of physics, and a field in the sense of abstract algebra. If the possible values of a physical field are drawn from some algebraic field (e.g. real or complex numbers), that is just a coincidence of names and has nothing to do with why the physical concept is called a 'field'.
Anyway, although the question focuses on the Higgs boson mass for some reason, I believe a more general question would be better:
"What are the consequences for quantum field theory, if the fields are ring-valued rather than field-valued?"
Though to answer this, perhaps one should first be able to answer an even more straightforward question:
"What properties of quantum field theory depend on the fields being field-valued?"
I don't have answers for any of these questions. They place QFT in a context which is unusually and perhaps even inappropriately generalized. The small generalization to quaternionic QFT is already exotic (but it has been studied); to ask about 'field-valued quantum fields' in general, is to ask about something that very few people would have even considered.
As for 'ring-valued quantum fields', the integers are a ring, and observables with an integer spectrum are commonplace in quantum mechanics; but perhaps the equation of motion would have to be a difference equation rather than a differential equation. In any case, integers are just one example of a ring.
To sum up, such a question appears to require a comparison between two classes of theoretical object that have never been studied. Perhaps some mathematician will take up the challenge. Or perhaps the ring-vs-field distinction can be used to draw some quick clever implication (e.g. with respect to the usual expectation that quantum operators will be hermitian or self-adjoint?).