Is there an interval notation for complex numbers?

Perhaps $\{z \in \mathbb{C}: \operatorname{Re}(z) \in [a,b], \; \operatorname{Im}(z) \in [c,d]\}$.

The complex numbers have no inherent order, so unless you invent something like $[[a+ci, b+di]]$ I know no more compact way to write this.

Maybe just define something as $[a,b]+[c,d]i$ ?

As far as I know there is no widely recognized standard notation. Define one in your paper/book/essay if you need it often.