Let me take a whack at this. The question in the wiki page is why the Minkowski signature is "privileged" and as usual one looks at the other possibilities, as did Synge in 'Special Relativity' p 17. Perhaps one says Privileged because experiment agrees with Minkowski. But if the world makes algebraic sense we should expand the search space to include Octonions and observe that complex quaternions are even subalgebras of complex octonions, and thus are Lorentz operators, which makes the signature of complex octonions +--- or -+++ (or some variant). I argue that the Minkowski signature is privileged because Complex Octonions are privileged, at least if one accepts that alternativity restricts the algebras to consider. Dixon does not - using OxHxC, which is not alternative, but is like SU(3)xSU(2)xU(1). [G.M. Dixon 'Division Algebras ... and the Algebraic Design of Physics'].
One feature indicating Complex octonions are priviliged is that one is not running thru signatures, but rather has a pair of opposite signatures, which happen to be Minkowski, so it is not put in by hand, or chosen to agree with experiment. The wiki page would appear to be missing something here. So Complex Octonions appear to Define rather than merely Accommodate spacetime, as would be the case with Clifford Algebra. In fact Clifford has +--- in a different algebra from -+++.
To push the case that Complex Octonions are 'really privileged' one would hope to see quanta defined as well as spacetime structure, but one might say the parentheses allow us to define duality oscillators, like the photon, and generalize to all quanta. Perhaps this is what is missing from the Standard Model.
Anyway, this is rather unlike Tegmark's Platonistic view, and no mention of anything Anthropic is needed. Minkowski doesn't just agree with experiment, it also 'makes algebraic sense'.
Back to your questions: Four generators looks like a natural number to me, and I can't make any sense of a 'negative amount of dimensions'. As to 10d etc, if the LHC finds supersymmetric boson-fermion pairing then experiment would invalidate the alternativity assumption above, and Complex Octonions would be only 'partly privileged'.