Complex numbers are a mathematical construct, used to represent two orthogonal sets of numbers named "real" and "imaginary". There are many orthogonal sets possible. For example, the sine and cosine functions are orthogonal to each other. This is useful because there are many situations in the real world that can only be adequately described by orthogonal function sets. Complex impedance is one of them, consisting of a "real" resistance and an "imaginary" reactance. In Cartesian coordinates, resistance is described by numbers on the X axis while reactance is described by numbers on the Y axis. The two numbers (x,y) are on the complex plane containing the X-axis and the Y-axis. The distance from the origin (0,0 to x,y) is defined to be impedance. Everything else is you study about complex numbers is just different ways of looking at the same thing mathematically.

Note there is nothing "imaginary" about reactance. A capacitor, or an inductor, does affect the current through itself when a voltage is applied. Or, conversely, it affects the voltage across itself when a current is applied. If these voltages and currents are oscillating sinusoidal waveforms of constant frequency, eventually a steady state is reached where the current waveform is shifted exactly ninety degrees in phase with respect to the voltage waveform. This always occurs, no matter what value of capacitance or inductance is involved. It is an intrinsic property of the electromagnetic storage of energy in capacitors and inductors. One way to describe this phenomenon is complex arithmetic.

Why do we "need" complex numbers? Well, we don't. I was calculating impedance while still attending grade school, using rote formulas, without much understanding of why the formulas worked. It was only much later, when things became more complicated, that the introduction of complex numbers actually simplified calculations. The more mathematics you learn, the more you learn what you don't know. There is no end to it, but the beauty of mathematics (a purely human construct of the mind!) is its application to describing the world around us. Some things cannot be adequately described. How do you describe colors to a person born blind? Until you can see the colors, no abstract explanation will suffice to explain what colors are. Same with complex numbers. Until you can see the mathematical beauty of orthogonal relationships, and how these relate to the world around you, complex numbers might as well all be imaginary.

BTW, orthogonal sets are not limited to just pairs. The Universe we move around in can be described by a three-dimensional orthogonal set, unless you move around in it really fast. Then things get really complex.