What is the complex dipole moment?

I have some problems with getting the complex (time dependent) dipole moments of some dipoles in a configuration. I eventually want to get the electric and magnetic fields of the configuration, but my biggest confusion is about the dipole moments. I wonder:

• How can you obtain the complex dipole moments mathematically?
• Does it mean anything physically when the dipole moment is complex? Or is it only used for simpler math?

I can give you an example:

Consider the following configuration. I want to calculate the complex dipole moments of the two dipoles (and later the electric field they are creating).

The dipoles have the dipole moments

$vec{p}_{1}=p_0cos(omega t)hat{z}$

$vec{p}_{2}=p_0cos(omega t-pi/2)hat{z}$

I would then use the Euler identity

$e^{ix}=cos(x)+isin(x)$

$cos(x)=frac{e^{ix}+e^{-ix}}{2}$

$sin(x)=frac{e^{ix}-e^{-ix}}{2i}$

which would give me the following dipole moments

$vec{p}_{1}=p_0frac{e^{iomega t}+e^{-iomega t}}{2}hat{z}$

$vec{p}_{2}=p_0frac{e^{i(omega t-pi/2)}-e^{-i(omega t -pi/2)}}{2}hat{z}$

However, according to my book the correct complex dipole moments should be

$vec{p}_{1}=p_0hat{z}$

$vec{p}_{2}=p_0e^{-ipi/2}hat{z}$

I can see that the author has removed the time dependent part, but I still think my result looks wrong. I feel quite confused about the complex dipole moments so I hope someone can make it a little bit clearer.