Complex rabi frequency

I am learning quantum optics.

In the book written by Scully includes complex rabi frequency, when dealing with $$\Delta$$ level configuration, the Hamiltonian including laser-atom interacting concludes complex rabi frequency.

However, from the course I took as undergraduate, where rabi frequency are treated as real numbers (vectors), I cannot see why rabi frequency has imaginary part.

Could anyone help me with this? Why it has imaginary part? And how to deduce it?

$$\Omega_{i,j} = \frac{\vec d_{i,j}\cdot \vec E_0}{\hbar}$$
where $$\vec d_{i,j}$$ is the transition dipole moment for the $$i\to j$$ transition and $$\vec E_{0}={\hat {\epsilon }}E_{0}$$ is the vector electric field amplitude. $$\hat\epsilon$$ is the polarization of the light, which can be complex (e.g. for circular polarized light). But I think more importantly, $$E_0$$ is in general complex: $$E_0 = \text{e}^{\text{i}\varphi}|E_0|$$.