They showed that no "real algebra"-valued Hilbert space works. That is literally not true. The abstract clearly states that they only compare Real and Complex-

They showed that no "real algebra"-valued Hilbert space works. That is literally not true. The abstract clearly states that they only compare Real and Complex-valued Hilbert Spaces; they say nothing about general R-algebras. And of course they couldn’t prove what you claim because C itself is an R-algebra! I’m not claiming that they’re cheating or that what they did is trivial, just that 1) I don’t think they’ve answered the very broad question “Does quantum mechanics need imaginary numbers?” because as stated it’s sort of ill-posed and unfalsifiable. What they have answered is “Can this Hilbert space formulation of quantum mechanics with Postulates 1-4 work with real coefficients?” and the answer is no. The article itself points out non-Hilbert formulations that do in fact avoid complex numbers. Maybe this narrow question is interesting to physicists, I don’t know! 2) I don’t personally think it’s an interesting question, because I don’t think complex numbers are anywhere near the top of the list of unintuitive abstractions needed for modern physics or mathematics. Square root of -1 is a lot easier to swallow than even Cauchy Sequences or Dedekind cuts, much less I dunno, topoi of presheaves on the category of commutative von Neumann algebras of bounded operators (I wish I were making that up: https://aip.scitation.org/doi/abs/10.1063/1.4898185 https://aip.scitation.org/doi/abs/10.1063/1.4898185).