We usually need 4 dimensions to graph complex functions where both the x and y axis have an imaginary axis, but I care more about $f\left(x\right)\in\mathbb{C}$ than $f\left(x\in\mathbb{C}\right)$.
With that being said, functions on a plane are not fit for complex numbers in the way that they require another axis to graph. It’s just a 2-dimensional cross section of a graph in which the z axis is the imaginary part of the y axis. Therefore, the function cuts off at zero. Luckily, we live in the third dimension and can graph this way. It’s a 3-dimensional curve. Of course, that’s also a 3D cross section of a 4D graph where the w axis is the imaginary part of the x axis.
The thing that I am asking has to do with how functions look with this method of graphing, since nobody else on the internet has answered or even asked this question. Functions I am especially wanting to look at are $x^x$, $\sqrt{x}$, $\sqrt[x]{x}$, and $\ln{x}$.