The idea of complex numbers took many years to become acceptable. They help fill in the gap left by the rule that that says we cannot take the square root of a negative number. The easiest way to think of complex numbers is as an ordered pair of numbers. We use the first as the horizontal movement and the second as the vertical movement. To look at fractals we need to be able to work with functions of these pairs of numbers.
Vector spaces are important structures in the study of linear algebra.
Notice that multiplication and division of pairs is not in this list. Ok, let's try some simple functions. Here is a list of functions. Those are correct. Let's start with a little history lesson.