In 3 dimensions
If we consider the real and imaginary parts of the function values separately, then a complex function can be thought of as a pair of \(\mathbb{R}^2 \rightarrow \mathbb{R}\) functions. And these can be independently plotted in 3D. (And then of course projected onto a plane.)
This procedure also leads to pretty images, the behaviour of the real and imaginary parts becomes easy to observe; but it is a bit cumbersome to mentally unite the two images in order to gain a full picture of the function.
The square function can be seen below. The formulas of the real and imaginary parts are deduced this way: \[ z^2 = (x+yi)^2 = \underbrace{x^2-y^2} + \underbrace{2xy} \cdot i,\qquad f_1(x,y) = x^2-y^2,\qquad f_2(x,y) = 2xy. \] We obtain a harmonic pair of functions.
\(f(z) = z^2\)
And the following pair of images depict the exponential function. The formulas \[ f_1(x,y) = e^x \cos\,y,\qquad f_2(x,y) = e^x \sin\,y \] give the partial functions.
\(f(z) = \exp\,z\)
We might indeed recognise the exponential function and also the sine and cosine functions.