For a more intuitive explanation, this paper on the advantages of complex wavelets switched on the lightbulb for me why signals are complex in the first place:
In signal processing we can take any periodic-in-time signal and pretend it is a function defined on a circle parametrized by angle measure.
Yes, but as always you can ask yourself: is this merely a coincidence (if there is such a thing as one) or is this reflecting some deeper relation between nature and mathematics?
After all, a "signal" is quite a general and abstract notion.