What do complex numbers mean physically?

Doesn’t work like that.

Let’s start with real numbers. What do they mean physically? Well, everything and nothing. Everything, because we use real numbers as models for position, distance, time, mass, temperature, energy, momentum, current and plenty of other things. Nothing, because none of these things is actually represented by a real number – we have no idea what goes on below a certain scale, and using real numbers as a model is just convenient.

Similarly, complex numbers don’t mean anything physically. We can choose to use complex numbers to model various things that nicely fit their behavior. For example, the impedance of certain electrical components like capacitors and inductors can conveniently be modeled with complex numbers, because their effect on periodic currents changes both magnitude and phases.

Is current, or impedance, “really” what a complex number is, or what it physically means? Not at all. Complex numbers are an idea which is decoupled from the physical world. But they can be used to model certain physical phenomena, like this one.

More importantly, complex numbers provide the basic framework for quantum mechanics – states, wave functions and so on. The way quantum mechanical systems evolve and interact with each other relies heavily on the algebra of complex numbers.

And still, it’s incorrect to say that the field of scalars of quantum theories is what complex numbers “mean”. They happen to be very suitable for that task, and a whole host of others, inside or outside the physical world.