The paper is describing a test of a particular alternative to quantum mechanics called "real quantum mechanics", its rather more specific than you are suggesting

https://arxiv.org/abs/2101.10873

I know a few of the people on that preprint, and I'm pretty certain they all know you can embed the complex numbers in the real 2x2 matrices ;)

As an aside I'm pretty sure the matrix you need is [[0,-1],[1,0]], which squares to -I, unless you don't want to use matrix multiplication in your construction.

I have read the introduction (only cursorily). What bothers me is that they may be conflating

C as C-vector space, which is R (as R-vector space) tensor C (complex dim 1).

with

R^2 as R-vector space, tensored by C (complex dim 2).

Because when they "add a real qubit", they seem to be adding a degree of freedom which you do not have when you have just one C-qubit.

But -I repeat- I have only looked over the intro.

This comment seems to have been cut off before the end?

Oh sorry. It was a repeat which I did not notice.

as I was writing that matrix out from memory I was thinking there was something funny... Thank you for correcting me.

No. You've just reinvented them. Writing something differently and giving it a different name doesn't do away with that something.

That's the point, in mathematics you can model the same object in many different ways save for trivial transformations.