For two continuous real-valued functions $(f,g),$ the convolution product is defined as
$$h(t)=(f*g) = \int_{-\infty}^{+\infty} f(\lambda )g(t-\lambda ) \, d\lambda $$
So for two complex valued functions $f(z)$ and $g(z)$ the convolution product would look something like:
$$h(z)=(f*g)=\int_{z\in \Gamma} f(\omega)g(z-\omega) \, d\omega$$
where $\Gamma$ is a closed contour containing singularities. Now if $f$ and $g$ are entire functions then $h(z) = 0$ so this would only be useful for meromorphic functions with isolated singularities.