About Hachel's alternate "complex" numbers

I did a bit of research to see if the structure that Hachel originally proposed, with these multiplication rules:  (a, b) * (a', b') = (aa' + bb', ab' + a'b)
had already been studied. Since it is clearly a ring (but not a field, as it has divisors of zero), it seemed likely to me.
And indeed, it has! This is called the set of split-complex numbers:  https://en.wikipedia.org/wiki/Split-complex_number
I came across it while watching a video by Michael Penn:  https://www.youtube.com/watch?v=r5mccK8mNw8
He demonstrates there that there are only three associative R-algebras over R^2:  - Dual numbers R(epsilon) with epsilon^2 = 0 (i.e. R[X]/(X^2))  - Complex numbers R(i) with i^2 = -1 \) (i.e. R[X]/(X^2 + 1)  - Split-complex numbers R(j) with j^2 = 1 (i.e.R[X]/(X^2 - 1))
Among these three, only the complex numbers form a field. All three also have a 2x2 matrix representation.
What should please Hachel is that split-complex numbers naturally express Lorentz transformations, since their isometries are hyperbolic rotations.
There is even an analogue to Euler’s identity:  e^(i*theta) = cos(theta) + i*sin(theta)
which is:  e^(j*theta) = cosh(theta) + j*sinh(theta)
However, note that while R(j) corresponds to Hachel’s *first* proposed structure, it has *nothing to do* with his *second* proposal of introducing an element such that (i^2 = i^4 = -1 ). As was pointed out to him (both here and on fr.sci.maths), this immediately leads to contradictions.  It is also completely absurd to claim, as he did before shifting to another incoherent idea (i^4 = i^2 = -1 ), that one of these three structures would be the "correct" one while the others are "wrong".  To make a long story short, Hachel had an idea that, for once, was not absurd. Strangely, he abandoned it in favor of another one that is completely incoherent and contradictory.