I think the point is that

a) The reals are most convenient for representing plausibility.

b) Any results that hold under the transitive and universal comparability laws will still apply if we represent plausibility with reals, so you might as well use them.

Complex numbers don't have an ordering, which seems counterintuitive as usually you can talk about things being more or less plausible. (Though they do get used in quantum mechanics for something that could be said to resemble plausibility, but that introduces a whole bunch of complicating factors that are not relevant here.)

When it comes to number systems with infinitesimals, you could often in principle use them, but in practice they aren't relevant because infinitesimal values will almost always be outweighted by non-infinitesimal values.

The thing that is special about $R$ is that it is the unique (up to relabelling) complete ordered Archimedean field.

You've already said that it has to be ordered. It also needs to be a field so we can add, multiply, and divide things corresponding to the various properties that arise from Cox's theorem. Completeness is exactly the "no holes" property.

The final one is the Archimedean property. This one says that for any element x>0 there is some natural number n such that $nx>1$. In terms of plausibility, it says that anything "infinitesimally small" in plausibility should be represented by zero, and not one of some infinite set of things that are for all finite purposes equivalent to zero.

If you relax the Archimedean property, you get systems such as hyperreals, surreals, and such. They all necessarily extend $R$, and so you do still end up needing to deal with $R$ anyway. They also have awkward properties when used as numbers by which to measure things (such as countable additivity failing to be sufficient, but uncountable additivity being too restrictive to work).

alternative number systems, which don’t have 1-to-1 correspondence to the reals
...
hyperreals or complex numbers