What is the “standard basis” for fields of complex numbers?

Just to be clear, by definition, a vector space always comes along with a field of scalars $F$. It’s common just to talk about a “vector space” and a “basis”; but if there is possible doubt about the field of scalars, it’s better to talk about a “vector space over $F$” and a “basis over $F$” (or an “$F$-vector space” and an “$F$-basis”).

Your example, $\mathbb{C}^2$, is a 2-dimensional vector space over $\mathbb{C}$, and the simplest choice of a $\mathbb{C}$-basis is $\{ (1,0), (0,1) \}$.

However, $\mathbb{C}^2$ is also a vector space over $\mathbb{R}$. When we view $\mathbb{C}^2$ as an $\mathbb{R}$-vector space, it has dimension 4, and the simplest choice of an $\mathbb{R}$-basis is $\{(1,0), (i,0), (0,1), (0,i)\}$.

Here’s another intersting example, though I’m pretty sure it’s not what you were asking about:

We can view $\mathbb{C}^2$ as a vector space over $\mathbb{Q}$. (You can work through the definition of a vector space to prove this is true.) As a $\mathbb{Q}$-vector space, $\mathbb{C}^2$ is infinite-dimensional, and you can’t write down any nice basis. (The existence of the $\mathbb{Q}$-basis depends on the axiom of choice.)