The study of algebraic geometry usually begins with the choice of a base field $k$. In practice, this is usually one of the prime fields $\mathbb{Q}$ or $\mathbb{F}_p$, or topological completions and algebraic extensions of these. One might call such fields $0$-dimensional. Then one could say that a field $K$ is $d$-dimensional if it has transcendence degree $d$ over a $0$-dimensional field.

But is there a way to make this less ad hoc? Is there a reason I've never seen any such definition of $0$-dimensional fields? Am I missing something?

To what extent has the classification of abstract fields been considered?