3D polarization possible, or is it always 2D: Horizontal, Vertical? A photon can be polarized vertically or horizontally (or linear combination of such):
$$
|\psi \rangle = \alpha |H \rangle + \beta | V \rangle
$$
Are there physical systems which can polarize in 3 dimensions:
$$
|\psi \rangle = \alpha |H \rangle + \beta | V \rangle + \gamma|D\rangle
$$
 A: For photons, you will never be able to have such polarisation:
$$
|\psi \rangle = \alpha |H \rangle + \beta | V \rangle + \gamma|D\rangle
$$
where $H$, $V$, and $D$ are linearly independent. Because the photon is a spin-1 massless gauge field $A^\mu$, its $4$ degrees of freedom are reduced to $2$ because of gauge fixing and masslessness. Massless spin-1 fields have 2 degrees of freedom and can hence only have $2$ independent polarisations, usually taken to be linear/vertical or right/left circular polarisation.
Light can still have a "longitudinal polarisation" in waveguides and in material, and by that one usually means that there is a component of the electric field $E_z$ along the direction of propagation $\hat{\mathbf{k}}$. In free space, Maxwell's equation gives you $\mathbf{E}\times\mathbf{B} \propto \hat{\mathbf{k}}$ so $\mathbf{E} \perp \hat{\mathbf{k}}$. But you still only have two linearly independent degrees of freedom.
On the other hand, spin-$1$ massive fields may have three linearly independent polarisations, or spin orientations. For example, $W^\pm$ and $Z^0$ bosons.
A: Yes, the third polarization is called longitudinal polarization. Here is a paper describing experimentally setting up transverse and longitudinal polarization.
https://cds.cern.ch/record/183186/files/SCAN-0008023.pdf
As you mentioned, this is not for light.
