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The question of "how big" is the cardinality of the continuum ($2^{\aleph_0}$) is rather tricky in set theory. It is consistent with ZFC that it could be bigger than naïvely expected (if the negation of the continuum hypothesis is true).

Apart from this, there are not (in my knowledge) fields of numbers of cardinality bigger than the continuum (even the hyperreals have such cardinality). And it is a rather unavoidable requirement of any sensible mathematical theory of QM that observables take values in a field of numbers, if else it would be very difficult (probably impossible) to match the mathematical predictions with the observations.

The question whether surreal or hyperreal numbers (that both contain the reals, even if they have the same cardinality) could be useful to provide a more satisfactory theory of QM is maybe more interesting. The mathematical evidences, such as the transfer principle for hyperreal numbers, suggest that probably a QM theory with hyperreal/surreal numbers would have essentially the same predictive power than standard QM as it is formulated, but would probably be more involved, and would have to be developed from scratch.

One may also think about developing a quantum theory in a different mathematical theory, mainly weakening the axiom of choice (that yields some counterintuitive results). For example, in the Solovay model (ZF+DC) every set of reals is Lebesgue measurable and $L^1$ and $L^\infty$ are duals of each other. The lack of AC for sets with large cardinality may however be rather inconvenient, especially since the algebra of observables satisfying the canonical commutation relations is, for example, non separable (and thus probably not much could be proved on it without the full AC). Nonetheless it may be worth to explore such directions, if not for immediate concrete applicability at least for the sake of knowledge.