Quaternions vs 6D Positive Spatial Math

I was trying to understand quaternions and i came with a new undertanding of a maybe improvement on rotational calculations.

I didnt mess up with quaternions calculations yet, but from what i understood, it looks like a 12 or double 6 order tensors calculations, correct?

if so, extinguishing the negative side of all axes, with a new axis with positive infinite, like, x,y,z,x’,y’,z’, like image 2. wouldnt be more performant? Because we could get the same rotational results with only a 6 order tensors.

I mean, on the compiler, changing the math operands removing the negative symbol (-) and replacing it again on the end of the calculus, by its ‘’'"(line version of the axis), would not be more performatic? Only having positive calculations and extinguishing complex numbers?

I apologize if I’m saying a nonsense, but appears to me a simpler way to improve language recognition than make and expensive calculus.