In the Chapter 7 of Jackson's book on Classical Electrodynamics, there's the following statement:

Introducing the complex orthogonal unit vectors:

$$\epsilon_{\pm}=\frac{1}{\sqrt{2}}(\epsilon_1\pm\epsilon_2)$$

we can represent an EM traveling in the $\epsilon_3$ direction with arbitrary polarization as:

$$\vec{E}(\vec{r},t)=(E_+\epsilon_++E_-\epsilon_-)e^{i(\vec{k}\cdot\vec{r}-\omega t)}$$

If the amplitudes have the ratio,

$$\frac{E_-}{E_+}=re^{i\alpha}$$

then it can be easily shown that the ratio of semimajor to semiminor axis is:

$$|\frac{1+r}{1-r}|$$

and that the ellise is rotated by an angle $(\alpha/2)$.

I tried checking this by myself, by starting with the general field $\vec{E}$ expression, passing to the basis with $(\epsilon_1,\epsilon_2)$ and expressing $E_\pm=A_\pm e^{i\beta_\pm}$, However, so far what I've found is that the $\vec{E}$ in such basis can be expressed as (taking the real components):

$$\vec{E_{1,R}}=\frac{1}{\sqrt{2}}[A_+Cos(\beta_++\vec{k}\cdot\vec{r}-\omega t)+A_-Cos(\beta_-+\vec{k}\cdot\vec{r}-\omega t)]$$

$$\vec{E_{2,R}}=\frac{1}{\sqrt{2}}[-A_+Sin(\beta_++\vec{k}\cdot\vec{r}-\omega t)+A_-Sin(\beta_-+\vec{k}\cdot\vec{r}-\omega t)]$$

which shows that we indeed have an ellipse. Nevertheless, since I can't seem to find the ratio and inclination from Jackson, and I couldn't find online a detailed derivation.