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In standard EM course, the complex EM field was emphasized and studied.

However, the complex dipole moment, such as time harmonic dipole moment $\vec{p} \exp{i\omega t}$ was less investigated, as most of the time complex EM field was taught in vacuum waves after the study of electric/magnetic multiple.

Some related posts could be found What is the complex dipole moment? and Complex charge of a Hertzian Dipole

What's the connection between complex charge, complex dipole, and complex electric field? Especially, is there any connection of complex dipole and complex charge with jone's calculus? i.e. polarization?

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In principle, one could use the complex notation to indicate an orientation of a dipole or the polarisation of an electric field. However, this is not what your question is implying. You specifically asked about the time varying complex notation $e^{i\omega t}$, so this is what I am going to address.

In this context the complex notation is merely a mathematical convenience. It helps us to perform calculations without invoking trigonometric relations. However, note that this mathematical convenience has its limitations: It is only allowed, if we perform linear operations, like multiplying with a scalar or adding several fields together.

While the complex notation is handy for doing the math, the physical quantities are always real numbers. Hence, we are actually only interested in either their real or their imaginary part -- which one we chose is arbitrary.

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