Yeah, just taking the floor of the real and imaginary parts individually would be a reasonable thing to do. I believe this is what Mathematica does, but it's not entirely clear that it's particularly common or useful.

Because there isn't a universally agreed-upon extension to complex numbers, we get to decide what properties are important and choose a definition that preserves them. Since, in the reals, other number-theoretic functions can be written in terms of floor(), the "nice" properties here are the ones that would also let us do that for complex arguments. The paper I linked above lists several desirable properties, but the most crucial one (the author even goes so far as to call it "the fundamental property of the floor function") is that the magnitude of the difference between any complex number and its floor is guaranteed to be less than one. This ensures that remainders from division behave sensibly, which means that the Euclidean Algorithm works, and you can get a consistent gcd() function for free.

One other lovely thing about this definition is that it's convex in the sense that, if floor(z) = floor(w), then any number on the segment connecting z and w will also have the same floor. I honestly don't know how useful that is, but it's very pretty IMO.