Where $D$ is the normal differential operator, $D f(x) = \frac{d}{dx} f(x)$, $D^n f(x) = (\frac{d}{dx})^n f(x)$, we can define, for example, the "half-derivative" $D^\frac{1}{2}$ such that $D^\frac{1}{2}D^\frac{1}{2} = D$.
If you work through the details, it turns out that, for real $\alpha$,
$$D^\alpha x^n = \frac{n!}{(n-\alpha)!}x^{n-\alpha}$$
In this video, Dr Peyam extends this to complex $\alpha$, constructing the "imaginary derivative" $D^i$.
At the end of this, he says he knows of no applications of this operator.
Can you think of any?