Can it be defined something like $D^{f(z)}$ operator?

Question

I've heard about fractional calculus where you can extend derivatives (and integrals) to complex numbers so that $\frac{d^{z}}{dx^{z}}\equiv D^z$ is meaningful where $z=a+bi$. So I was wondering if it is possible to definde something as $D^{f(z)}$ where $f(z)$ is a complex function