I never truly understood complex numbers until I had to take signal processing in my sophomore year. I had held a technician class-equivalent ham radio license since I was 12 but digital signal processing was pretty much nonexistent in the exams and training material.

It was common knowledge you had to work with filtering and intermediate frequencies because negative frequencies would be reflected along the 0 Hz axis and overlaid on the positive frequencies but mirrored. With complex numbers you suddenly were able to downconvert signals directly to the baseband and keep negative frequencies separate.