Enforcing the Unity of Space and Time Using Quaternions

Presented to the Fifth International Conference on the Ontology of Space-time in Albena, Bulgaria, May 2018. A view of number theory that can be applied broadly to physics.

Table ot contents
3:04 Zero, one, and the trivial group
10:00 Real numbers
13:56 Complex numbers
19:39 Space-time numbers
25:04 Simple math = physics
26:21 Special relativity & quaternion gravity
36:01 Purple Penguin derivation
48:00 Quaternion gravity and EM
50:55 Quaternion series QM
56:01 Testing quaternion series QM
59:45 Summary

[purplepenguin]

I noticed recently that while there are proofs that the only finite-dimensional division algebras over the reals are of dimension 1, 2, 4, or 8, there is no proof that there are not "weird" division algebras besides the usual R, C, H, and O.
I thought for sure the 2 dimensional case would be constrained enough that complex numbers would be the only possibility, so I started there... but instead of proving complex numbers were the end of the line, I seem to have found something weird.

The multiplication of an algebra must be bilinear, so if I assume there is a left identity the 2D case is fairly constrained. So I looked at some of the options and found that (a, b)(c, d) = (ac+bd, ad-bc) appears to be a division algebra without a right identity. Its like having an extra "sqrt(1)" that anti-commutes with real numbers. Weird.

The inverse of left multiplication is given by (c/(c^2+d^2), d/(c^2+d^2))(c, d) = (1, 0)
Which in turn hints at a norm: (a, b)(a, b) = (a^2+b^2, 0) ... Yet there is a proof complex numbers are the only 2D normed real division algebra. So either I'm making silly mistakes, or I've been misunderstanding what those proof really say. I'm hoping the former. Do you see my mistake?

If instead this is a known real thing, any idea what the name of this is?

[purplepenguin]

Random thought that popped in my head today.
Octonians every once in awhile get some attention from physicists. But they are so strange. They don't even have a matrix representation because they are non-associative. So what if ... there was an even stranger number extension that no one thought of because even addition becomes non-commutative or non-associative? I can't even figure out what that would even mean. Since you love exploring number systems, how would you try to explore that and try to come up with definitions to study the properties?

Anyway, I hope the summer is treating you well.