Since there are complex numbers (2 dimensions) and quaternions (4 dimensions), it follows intuitively that there ought to be something in between for 3 dimensions (“triernions”).
Yet no one uses these. Why is this?
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 R, 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.