The fact that complex numbers allow you to get a much more accurate approximation of the derivative than classical finite difference at almost no extra cost under suitable conditions while also suffering way less from roundoff errors when implemented in finite precision:
What?
The formula you linked is wrong, it should be O(epsilon). It’s the same as for real numbers, f(x+h) = f(x) + hf’(x) + O(h^(2)). If we assume f(x) is real for real x, then taking imaginary parts, im(f(x+ih)) = 0 + im(ihf’(x)) + O(h^(2)) = hf’(x)) + O(h^(2)).