Suppose $f:D \subset \mathbb{R}^2 \to \mathbb{R}^2$.
If $f$ is differentiable (or $\mathbb{R}$-differentiable) we say $f \in C^1(D;\mathbb{R}^2)$ or $f \in C^1(D)$ (meaning the partial derivatives exist and they are continuous).
Now if $f:D \subset \mathbb{C} \to \mathbb{C}$, is it correct to say that $f \in C^1(D;\mathbb{C})$ or $f \in C^1(D)$ if $f$ is $\mathbb{C}$-differentiable (meaning it is $\mathbb{R}$-differentiable as a vector field and the Cauchy-Riemann equations are satisfied).
If it is incorrect, what is the correct symbol for a $\mathbb{C}$-differentiable function $f$?