If we start with the proven fact light travels at the same speed in all reference frames, and add the proven fact a difference in speed causes a red or blue shift in frequency. Then it seems like doppler shifts in light do not work the same as sound.
It's my understanding, and could be wrong, that sound experiences a doppler shift because the source's differing speed relative to sound can be different. An oncoming object piles the sound up a bit compressing it's wave. Away going objects stretch it. All because their relative speed of sound through air is different. Sonic booms happen when an object outruns it's own sound.
However light is always the same speed relative to whatever emitted it, it can't pile up like sound. An object going 0.999999 the speed of C, and emitting a photon will see the photon race ahead at 1C and so will an object moving 0.000001 C. So what I'm wondering about is how doppler shifts work in light. My best guess is a blue shifted objected has a different relative clock than mine. From my perspective it's experiencing time faster, so any light it emits is going to have a higher frequency than it would other wise because any clocks it would have would tick faster, and frequency is a function of time. Red shifted objects would have a slower clock.
Is this correct or what's the correct way to conceptualize red shift and blue shift?