There are plenty of visual animations explaining what convolution and serial correlation are for real values.
However, for complex numbers, it is interesting that convolution does not involve conjugation, while serial correlation does. It is also unclear how the geometry of convolution and serial correlation would work in complex space generally.
Are there any animations available showing these operations over the complex plane?
I cannot provide any useful link to any animations. Here is a verbal description of the geometry of complex convolution that might be helpful.
Recall: For real-valued functions in which $f\geq 0$, the convolution $f*g =\int f(t-s) g(s) ds$ can be visualized as a weighted moving average of the values of $g$ using the weighting function $f$.
If $f$ and $g$ are complex, you can factor $f(t)= F(t) U(t)$ where $F(t)= |f(t)|$e $U(t)$ takes values on the unit circle. Multiplication by such a factor $U(t)$ performs rotations. Thus $U(t-s)) g(s)$ describes a rotated collection of values of $g(s)$. Next we apply the factor $F(t)$ and integrate to obtain a weighted moving average of the rotated version of $g(s)$.
In a computer animation I would depict the signal $g(s)$ as vectors, as a collection of spokes of variable length $|g(s)|$ pointing out from the $s$ axis. Then spin these spokes using the factor $U(t)$ and take the weighted average of the pattern so obtained.