Beginner's Question: What does a complex power mean?

When we get the output power of some device to be a complex number, physically what does it mean? what does the real and imaginary part mean respectively? And what is the actual power that can be measured? Thanks for your answering!

Real = Energy/power is consumed (Resistors heating up, Amps Amplifying)

Imaginary = Energy is stored in a field (Electric ,CAP, or Magnetic ,Inductor), thus the
energy is not used or wasted (Resistor dissipation of energy). When the field collapses
the energy goes back to the system or source.

Complex numbers are used extensively in electrical engineering for methematical convience they provide in analysis of elctrical systems. power can be taken as a complex number having an amplitude and a phase. if you take the real component of this number then this is the power actually being used by the circuit. Imaginary component is not power in strict sense , since volatage and current are 90 degrees out of phase, therefore no power is actually consumed by circuit but the circuit keeps storing and returning the power. This component however is important since the amplitude of current in the circuit depends upon the complex power and not just effective (real power). however, physically imaginary power has no significance of its own.

if you connect a wattmeter to circuit, it will measure the effective power only. the complex power is measure by measuring the current and voltages and relative phase between the two.

Then for example, if I know the power of a transmitting antenna is P=A+jB, then A is actually related to the energy dissipated in the form of heat (or something not contributing for creating the EM field), and B is related to the energy resides in the EM wave created by this antenna, Is that right?
I think in most cases, we care more about the magnitude of the power instead of the real and img part of it, why is that? Could you explain this more detailed from antenna point of view? Thanks a lot!

shouldn't it be the magnitude of |A+jB| which gives you the real power? (P[real]=sqrt(A^2+B^2))
arctan (B/A) gives you the phase.