A quick search here shows me this question: [No extension to complex numbers?](https://math.stackexchange.com/questions/142330/no-extension-to-complex-numbers) where they explain there's no 3D or 4D analogue of complex numbers. Then there's a mention of quaternions but the multiplication is non-commutative. This has left me a bit "confused."

Anyway, I just learnt a theorem that if $F$ is field and $p(x) \in F[x]$ is an irreducible polynomial of degree $n$ then $K=F[x]/(p(x))$ is an $n$-dimensional vector space over $F$. 

So what if $F=\mathbb{R}$ and $p(x)=x^4+1$, then $\mathbb{C} \subset K$ and $[K:\mathbb{R}]=4$, right?

Specifically, if $\theta=x + (p(x))$ then $\{1, \theta, \theta^2, \theta^3\}$ are the basis for $K$ and $\mathbb{C} \cong \{a+b\theta^2 : a,b \in \mathbb{R}\}$.