All correct interpretations of the Doppler effect imply that the speed of light relative to the observer varies with the speed of the observer. Sometimes this is stated explicitly - if the observer starts moving toward the source with speed v, the speed of the light waves relative to him becomes c'=c+v:

http://a-levelphysicstutor.com/wav-doppler.php
"Vo is the velocity of an observer moving towards the source. This velocity is independent of the motion of the source. Hence, the velocity of waves relative to the observer is c + Vo. [...] The motion of an observer does not alter the wavelength. The increase in frequency is a result of the observer encountering more wavelengths in a given time."

http://physics.bu.edu/~redner/211-sp...9_doppler.html
"Let's say you, the observer, now move toward the source with velocity Vo. You encounter more waves per unit time than you did before. Relative to you, the waves travel at a higher speed: V'=V+Vo. The frequency of the waves you detect is higher, and is given by: f'=V'/λ=(V+Vo)/λ."

Pentcho Valev