QFT Hilbert spaces over other rings than the complex numbers $\mathbb{C}$

Actually, all that is quite known from the foundational work by von Neumann and Birkhoff. In this formulation of QM (and in the subsequent evolution of this research area) one constructs the quantum theory theory starting from the lattice of elementary "YES-NO" observables (see my answer on quantum probabilities for more details) or "elemetary propositions" experimentally testable on a quantum system. These assumptions describe the common phenomenology of all quantum systems. That lattice turns out to be $\sigma$-complete, orthomodular, separable, atomic, irreducible and verifying the so called "covering property". In the standard QM this lattice is the one of orthogonal projectors in a complex Hilbert space. However already von Neumann noticed that at least two other possibilities seemed feasible in principle: the lattice of orthogonal projectors in a real Hilbert space and the lattice of orthogonal projectors in a quaternionic Hilbert space. In all cases, states are generalized probability measures on the relevant lattice.

This idea remained a long standing conjecture till 1995, when Solér (e.g. have a look at this entry of Stanford Encyclopedia of Philosophy) proved von Neumann's conjecture ruling out other formulations on different Hilbert space-like structures (e.g. using Clifford algebras as space of scalars). It rules out also Hilbert spaces constructed on the non associative algebra of octonions in apparent contradiction to what stated in the paper you mention.

In the presence of a time reversal operation, real quantum mechanics can be proved to be equivalent to the standard complex version. Instead the quaternionic one could contain some new physics, at least it is the opinion of S. Adler who wrote a thick book on this idea from a very physical viewpoint.

Some of fundamental theoretical results in QM survive the passage to quaternionic QM, like Wigner, Kadison, Gleason theorems (the last one is fundamental as it proves that the states are nothing but densitiy matrices and vector states as assumed in more elementary formulations of QM). Varadarajan's book on the geometry of QM deals with the three formulations simultaneously.

Quaternionic QM involves an interesting noncommutative functional analysis form the pure mathematical viewpoint (see for instance this paper of mine).

For these reasons I do not think that the paper you mention - which assumes to deal with some sort of Hilbert space whose scalars are elements of a Clifford algebra - may present a theory consistent with the basic standard assumptions of quantum theories formulated in Hilbert spaces or generalizations. This is just in view of Sòler's theorem. However the paper is not written in that clear mathematical fashion as the subject would deserve, in my honest opinion, for so mathematically, physically, and philosophically delicate issues.

To be honest I might also say that Hilbert space formulation is not the only possible. A more recent and in a sense more powerful is the algebraic formulation where the fundamental objects are not elements of a lattice and generalized probability measures on that lattice, but they are elements of a unital $C^*$-algebra (or more weakly a $*$-algebra or a Jordan algebra) the Hermitian ones representing the observables of the system. States are now defined by normalized positive functionals on the algebra, representing expectation values. The celebrated GNS reconstruction theorem proves that, when a reference state is chosen, this algebraic picture is equivalent to a standard construction - à la von Neumann say - in a Hilbert space. There are however many unitarily inequivalent Hilbert space realizations of the same algebraic structure.

However, the paper you mention does not seem to deal with this more abstract formulation.

(Regarding Stefan Hollands-Bob Wald's paper, I know quite well the authors and the ideas contained in that paper and I discussed them with Stefan in the past. I cannot see well how these ideas have much to do with alternative formulations of quantum theories at the level of the question I am answering. The point, there, was the re-formulation of quantum field theory, avoiding the standard perturbative approach. As far as I remember the basic structure of the Hilbert space does not play any fundamental role.)

Ok, to expand my comment into an answer:

There are only two finite-dimensional division rings (that admit division) containing the real numbers as a finite subring: the complex numbers and the quaternions (application of Frobenius theorem). Also, a vector space (and Hilbert spaces are vector spaces) is usually defined over a field, that is a non-zero commutative ring. Over a ring you have the module that is a generalization of a vector space on non-commutative rings (as the quaternions, and the supposed Dirac matrix algebra).

So even supposing you can generalize a Hilbert space as a module over a ring with a scalar product (I am not sure it is possible); the only way to allow for division, and to interpret suitable scalar products as a real probability, is to use either reals, complex (usual Hilbert spaces), quaternions or infinite dimensional division rings. Since it does not seem to be the case here, I suppose: either he is referring to quaternions in some sense, or he is mistaken, or he has to redefine from scratch the concept of probabilities, Hilbert space etc...