David Schneider-Joseph writes on Facebook:

In principle, this isn’t really any more mysterious than, say, -1/2 showing up somewhere. It’s unfortunate that imaginary numbers got called imaginary because they’re not really any less real than “real” numbers.

Numbers, even “simple” ones like the positive integers, are abstractions that we use when they map well onto reality. We never find the number 3 out there in the physical world. We may find three apples or a representation of the number 3, yet 3 itself is not to be found. But this is no problem: 3 is a concept that helps us make sense of situations in which we have three of something.

For someone used to using numbers to count things, the concept of a negative or fractional number seems sort of … imaginary. How can you have a negative number of apples? How can you have half a dog? The answer is that those sorts of numbers are designed for different situations: e.g., debts or backward motion in the case of negative numbers, divisible things like gallons or pizza pies in the case of fractions.

Similarly, complex numbers are just designed for a new sort of thing. Their algebra maps naturally to, among other situations, the behavior of periodic systems such as a wave (including very classical waves such as sound waves and vibrating strings), the magnitude and argument of the number representing the amplitude and phase of the wave. It just means an amplitude of 1 and a 90° angle from the starting point.

(I did some minor spelling corrections)