Why are There No “Triernions” (3-dimensional analogue of complex numbers / quaternions)? [duplicate]

It’s because there isn’t one! (Indeed, Hamilton was originally searching for such a thing, and found the quaternions instead; it was only later that people understood why he hadn’t been successful, initially.)

The quaternions – along with the real numbers and the complex numbers – have a number of nice properties: specifically, they form a real division algebra. This is a mouthful, but basically amounts to:

• Addition/multiplication of quaternions satisfy the ring axioms.

• We can divide by quaternions.

• We can multiply a quaternion by a real (and this “scalar multiplication” satisfies the basic properties it should).

It turns out the only finite-dimensional real division algebras are $\mathbb{R}$, $\mathbb{C}$, and the quaternions; see this. (I include associativity in the definition of algebra: if we allow non-associative algebras, then the octonions also count.)

By the way, there is a way to (sort of) keep going past the quaternions: the Cayley-Dickson construction. This produces things like the octonions and the sedenions, and other delightfully weird algebraic structures. However, it has a couple drawbacks:

• Each time you apply Cayley-Dickson, the dimension of the starting algebra doubles. So this won’t help us get to $3$.

• Also, you keep losing nice properties. Passing from the reals to the complex numbers, we lose order; going from the complexes to the quaternions, we lose commutativity of multiplication. If we keep going, we lose associativity of multiplication, in increasing degrees: the sedenions are even less associative than the octonions, etc.