Do photons age in a medium? According to special relativity, time starts to slow down as we increase our speed and eventually stops once we get to the speed of light. By that logic, photons don't age in a vacuum state as, to us, the time stops for them. However, in a medium, their speed decreases, that means time is 'stationary'. Does that mean they start to age in a medium?
 A: I will turn my comment into an answer, because the question in the header:

Do photons age?

is very anthropomorphic , and physics is a discipline that discourages interpreting data by use of  the anthropic principle.
The photon is an elementary particle. Aging is not a verb to be used with elementary particles in general because
a) they have no identification other than their quantum numbers and thus cannot be tagged to be checked, 
and
b) the quantum numbers associated with them like spin, parity, intrinsic angular momentum are invariant in time 
as also 
c) is their mass. 
So whether one is talking of photons or the other elementary particles in the Standard Model, the answer is the same:no meaning can be attached to the verb "age" with respect to elementary particles
A: Photons don't have a rest frame, since in all inertial frames they must go at the speed of light.
So the following statement:

By that logic, photons don't age in a vacuum state as, to us, the time stops for them.

is meaningless because one really can't talk about proper time for a photon.

However, in a medium, their speed decreases,

Nope. The net speed of field propagation decreases. However, the photons still move at $c$. When light travels through a medium, it induces electric field vibrations that emit photons in other directions, changing the net field velocity without changing the velocity of the photons.
