I depends on the book you've chosen to read. But usually some basics in Calculus, Linear Algebra, Differential equations and Probability theory is enough. For example, if you start with Griffiths' Introduction to Quantum Mechanics, the author kindly provides you with the review of Linear Algebra in the Appendix as well as with some basic tips on probability theory in the beginning of the first Chapter. In order to solve Schrödinger equation (which is (partial) differential equation) you, of course, need to know the basics of Differential equations. Also, some special functions (like Legendre polynomials, Spherical Harmonics, etc) will pop up in due course. But, again, in introductory book, such as Griffiths' book, these things are explained in detail, so there should be no problems for you if you're careful reader. This book is one of the best to start with.
I will expand my comment above into an answer:
If you search for a TOE that is a mathematical theory, it has to be at least a logic theory, i.e. you need to define the symbols and statements, and the inference rules to write new (true) sentences from the axioms. Obviously you would need to add mathematical structures by means of additional axioms, symbols, etc. to obtain a sufficient predictive power to answer physically relevant questions.
Then, given two theories, you have the following logical definition of equivalence:
let $A$ and $B$ be two theories. Then $A$ is equivalent to $B$ if: for every statement $a$ of both $A$ and $B$, $a$ is provable in $A$ $\Leftrightarrow$ $a$ is provable in $B$.
Obviously you may be able to write statements in $A$ that are not in $B$ or vice-versa, if the objects and symbols of $A$ and $B$ are not the same. But let's suppose (for simplicity) that the symbols of $A$ and $B$ coincide, as the rules of inference, and they only differ for the objects (in the sense that $A$ may contain more objects than $B$ or vice versa) and axioms.
In this context, ZFC and Bernays-Godel set theories are equivalent, when considering statements about sets, even if the axioms are different and the Bernays-Godel theory defines classes as mathematical objects, while ZFC does not.
Let's start to talk about physics, and TOE, following the discussion in the comments. It has been said that two TOEs must differ only in non-physical statements, since they have to be TOEs after all, and thus explain every physical observation in the same way. I agree, and from now on let's consider only theories in which the physical statements are true.
Let $A$ be a TOE. Let $a$ be an axiom that is independent of the axioms of $A$ (that means, roughly speaking, that there are statements undecidable in $A$, that are decidable in $A+a$, but all statements true in $A$ are still true in $A+a$). First of all, such an $a$ exists by Godel's theorem, as there is always an undecidable statement, given a logical theory. Also, $a$ is unphysical, since $A$ is a TOE. Finally, $A$ and $A+a$ are inequivalent (in the sense above), and are TOEs.
One example is, in my opinion, the generalized continuum hypothesis (GCH): without entering into details, it has been shown with the theory of forcing that it is independent of the axioms of ZFC set theory. Thus $ZFC$, $ZFC+GCH$ and $ZFC+\overline{GCH}$ (ZFC plus the negation of GCH) are all inequivalent theories that contain $ZFC$. It is very likely that a TOE must contain set theory, e.g. ZFC. Let $A$ be such a TOE. Also, it is very likely that $GCH$ is not a physically relevant axiom (at least it is not for our present knowledge). Then $A$ and $A+GCH$ would be inequivalent TOEs, then a TOE is not unique.
I have studied a bit of logic just for fun, so I may be wrong...If someone thinks so and can correct me is welcome ;-)
Best Answer
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.