Do complex numbers really exist? Complex numbers involve the square root of negative one, and most non-mathematicians find it hard to accept that such a number is meaningful. In contrast, they feel that real numbers have an obvious and intuitive meaning. What's the best way to explain to a non-mathematician that complex numbers are necessary and meaningful, in the same way that real numbers are?
This is not a Platonic question about the reality of mathematics, or whether abstractions are as real as physical entities, but an attempt to bridge a comprehension gap that many people experience when encountering complex numbers for the first time. The wording, although provocative, is deliberately designed to match the way that many people actually ask this question.

Erase from your mind that 'complex' numbers have anything to do with the square root of $-1$. Instead ask yourself if this is feasible:
Intelligent life in a distant part of the universe developed in a completely different way than we did. They often say
God made the polygons, all else is the work of ___.
This species made great strides in algebra at a very early stage. They built up considerable 'mathematical potential energy' in their ancient history.
In a short span of time, they went from the polygons to the circle, and then realized that there was a natural covering map of the 1-dim number line onto the circle group. For them, the circle group of rotations was just as useful as the number line.
As they worked with 2-dim space, one of their greatest minds realized that the most natural way to extend multiplication from the circle and number lines to all the points in the plane was to multiply number lengths and add their angles (dilation/rotation). It was a flash of intuition and insight, but the theory details gave then a tremendous technological advancement; many math problems they worked on had a new feel to them (for example, they found that every polynomial could now be completely factored).
As they advanced into quantum mechanics, they developed, what we humans call, the Schrödinger equation,
$i\hbar {\frac {\partial }{\partial t}}\Psi (\mathbf {r} ,t)={\hat {H}}\Psi (\mathbf {r} ,t)$
Of course they did not express it as we do, and they were really surprised when they learned that $i$ meant "imaginary number" to so many people on earth. They were even more surprised to learn what percentage of people even knew what it stood for.
