> I'm not good enough at math/geometry to know if this kind of 'next level up' bifurcation of dimensionality extends up past Quaternions or not (like something called Octernions, 16ions, 32ions, 64ions, etc), but it seems like is would?
Octonions and up (more generally known as hypercomplex numbers) exist, but every time you pull the "double dimensions by adding more imaginary components" trick[0], you lose another useful property.
Real to complex loses total ordering. Complex to quaternion loses commutativity. Quaternion to octonion loses associativity (but they are at least alternative). The sedenions aren't even alternative, and they have zero divisors to boot.
You can also generalize hypercomplex numbers to the study of Clifford algebras.
Octonions and up (more generally known as hypercomplex numbers) exist, but every time you pull the "double dimensions by adding more imaginary components" trick[0], you lose another useful property.
Real to complex loses total ordering. Complex to quaternion loses commutativity. Quaternion to octonion loses associativity (but they are at least alternative). The sedenions aren't even alternative, and they have zero divisors to boot.
You can also generalize hypercomplex numbers to the study of Clifford algebras.
[0] The Cayley-Dickson construction