Photons always travel at the speed of light in every frame of reference. Only the medium matters, not the observer's speed.The observable change is the separation between photons (wavelength) caused by a moving source.

 Photons traveling along an X axis are traveling along a Y axis at 0 m/s. If you're traveling along the Y axis, you're going faster.
 Except that's not what we measure. Instead we find the speed of a photon depends only on the medium, not on the motion of the source or receiver.
 You aren't listening to me. The speed of the photon is c in the direction it's traveling, a photon traveling due west is going north with no speed at all.
 I don't think it makes sense to say that the speed of a photon is c "in a direction", since speed is a scalar quantity.(correct me if I'm wrong, I'm terrible at physics!)
 Here, speed is a measure of the magnitude of velocity. It's perfectly acceptable to describe velocity (or a component thereof) as speed in a particular direction.

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