Because the "speed of light" is an upper limit on velocity, you can't add the speed of the source emitting light (the galaxy in your example) to the speed of the light itself and get a faster speed. Instead what happens is that the light coming from the approaching galaxy appears compressed in wavelength (that is, blueshifted) compared to if the galaxy were standing still relative to us. Spectrographic studies of galaxies show this effect. In some cases the rotation of a galaxy around its own center will result in blueshifted light from the part rotating toward us and redshifted light from the part rotating away from us.
-Dave-
Yep, and the fact that we can see the galaxy side-to-side, regardless of which direction it rotates, tells us something else: the photons emitted are always emitted at the speed of light. Their speed towards us is not influenced by the speedvector of the source. Even if the source is moving away from us, you don't have to subtract that speed. The photon still travels with maximum lightspeed towards us after emission. Which perhaps isn't as strange as it seems since they don't have mass and therefore the speed of the source isn't transferred to the photon by impuls or other conventional earth physics.
So, the questions becomes:
1. how is the photon created
2. what causes the photon to be compressed/expanded?
The problem of course is, this being relativistic, that neither the source, nor the observer is moving as far as the photon is concerned.
Something happens during the exact moment of creation that influences the (perception of) the photon.
Some possibilities:
If the duration of creation of a photon is always fixed, then the movement of the galaxy relative to me changes that duration in my perception. So that could cause a shift. Though this doesn't quite explain why there is such a thing as gravitational shift.
If the creation of a photon is influenced by the size of an electron jump (classical model) then perhaps the electrons are moving in a non-circular cloud around their atom core. This would happen in a gravitation field, or when the atom has been under acceleration.
Whatever happens during creation, the same thing should happen in reverse during the detection of the photon. If both me and the galaxy are moving at considerable speed in the same direction, I would not see any shifts. In fact, I would be unable to tell we are both moving.