Nonsense. Just use a [[1 0] [0 -1]] matrix and it's kin and do away with "essential" imaginary numbers.
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.
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.