You don't need to think in 4-D, just put two complex planes, one the domain the other the codomain, next to each other.
To get an idea of the action of the map you can begin by plotting points in the domain and their images in the codomain.
Next you can put in all real numbers and see what is done to that, or the imaginary numbers (they get mapped onto the unit circle.
If you want to know what happens to your other discs parametrize the boundary circle, say $z=a\exp(it)$ with $0\le t\le2\pi$, and calculate the image curve; you'll find it's another circle.