This was one of the points of contention with the quantum revolution of the beginning of the 1900's, schrödinger came up with the equation, which fitted like a glove for a lot of scenarios, but it had an imaginary component, which baffled a lot of people since it could imply reality uses such numbers at a fundamental level
What's really screwy is you can force light to only travel as a evanescent wave. It's completely undetectable without a second interaction, but light must transmit energy using the purely imaginary part of the complex wave.
The imaginary component definitely has some physical meaning, it's not just a useful mathematical trick.
This is why some observers have noticed that religion, 'the God of the gaps' especially, is dying and losing any use or meaning, leading to less metaphysical thought in everyday life, that math and physics especially now use metaphysical thought is the primary tool of new understanding and discovery. Which is bringing it back into everyday life.