On teaching complex numbers

Complex numbers are usually introduced to high-school students in a rather confusing manner. Till a certain grade they are told that the square-root of a negative real number is “forbidden”, and then is introduced all of a sudden. It is a pity that despite being able to solve a ton of problems, most students cannot perceive complex numbers beyond mere algebraic manipulations. The true potential of complex numbers is perhaps better understood through a geometric viewpoint, followed by its applications in other topics such as geometry, trigonometry, polynomials, combinatorics, and so on.

Different domains-different story

Do there exist functions such that for all ?