It is difficult to know where to take this question.
Correct me if I am wrong but you seem to be asking for a physical manifestation of the imaginary part of a complex number.
There are solutions to the schrodinger equation where this necessary but you need to know a lot of quantum mechanics to do this.
So I am trying to help you find another area of physics where you might find it easier.
That is why I am concentrating on alternating electric circuits.
If I asked you to tell me the current flowing in an inductance you could give me a number, say 5 amps and we would both agree that this current had real world existence.
However
That would not be enough to calculate the power since multiplying by the voltage would not give this correctly.
The question of when is the current needs also to be answered.
This information can be contained in another (ie a second) real number or it can be combined into one single complex number, which is much easier.
There are many situations where two pieces of real world physical infomation must be manipulated together to obtain a valid physical answer.
Complex numbers are one way of doing this, vectors are another (they are not the same).
But that is the easy stuff.
The stress tensor has 9 separate pieces of information to manipulate together to fully descibe a 3D situation.
FYI another big area in the use of complex numbers in physics is in conformal mapping to plot fluid flows, electric and magnetic fields and other contours in potential theory.
