Also a PhD in math, where complex numbers are fundamental, and also part of large swaths of similar structures that are also fundamental. They fit in nicely among a ton of other similar structures and concepts, so they seem about as fundamental as sets or addition or groups or fields (and there it is).
They also seem fundamental to physical reality in a way most math concepts do not: they're required (in structure) for quantum mechanics, in many equations that seem to be part of the universe. The behavior of subatomic particles (and more precisely, QFTs), require the waveforms to evolve as complex valued functions, where the probability of an event is the magnitude of the complex value.
This has been tested between theory and experiment to about 14 decimal digits precision for QED.
I'd guess they should be considered as real as radio waves (which we don't see), as the fact things we think are solid are mostly empty space (which we don't feel), or that time flows at different rates under different situations (which we also don't experience). Yet all those things are more real than stuff our limited senses experiences.
They also seem fundamental to physical reality in a way most math concepts do not: they're required (in structure) for quantum mechanics, in many equations that seem to be part of the universe. The behavior of subatomic particles (and more precisely, QFTs), require the waveforms to evolve as complex valued functions, where the probability of an event is the magnitude of the complex value.
This has been tested between theory and experiment to about 14 decimal digits precision for QED.
I'd guess they should be considered as real as radio waves (which we don't see), as the fact things we think are solid are mostly empty space (which we don't feel), or that time flows at different rates under different situations (which we also don't experience). Yet all those things are more real than stuff our limited senses experiences.
There's some string of research on if/how fundamental complex numbers are to QM, e.g., https://www.scientificamerican.com/article/quantum-physics-f...