The idea of the 'colorful visualization'
The main concept of the 'colorful' method (domain coloring) is as follows.
First color the complex plane somehow. (I.e. assign a—possibly unique—color to each complex number.) And then plot a function by painting each point of a (e.g. square) domain of the complex plane the color assigned to the function value at that point. (Of course in practice we examine not every but just enough points.)
Thus we get a colored square. (We shall see how concise, perspicuous and beautiful images are acquired this way, enabling us to directly read many different properties of complex functions. This method could give us a new insight, by which we may also gain deeper understanding of these functions at hand. But with respect to printing: this procedure is not very ink-friendly.)
Yet we haven't said anything about how to color the complex plane. One possible method would act so: let a color assigned to a complex number contain the more red the greater the real part is and the more blue the greater the imaginary component. This coloring is illustrated on the figure below. (Let us call this coloring Imre because it treats the real and imaginary parts separately. Note that 'Imre' is also a valid hungarian first name.)
A coloring (Imre).
So once more. Given a coloring (above), a function to plot and a selected domain (below) on which we intent to create the plot: For each point in our domain, paint it the color assigned to the function value in that point—one could get this from the coloring.
A complex domain, still unfilled.
Note that we already have the plot of the complex identity function, i.e. f(z) = z. (With the Imre coloring, to be specific.)