Nope, unlike the fly in the aircraft, both the "moving" observor, on the spaceship, and the "stationary" observor, will measure the light to be moving 186,000 miles/second (3E8 meters/second). That is both Einstiens assumption in developing the equations of special relativity, and obervationally verified. As are at least some of the predictions of SR, such as time running slower for moving objects. (observed by measuring decay times of radioactive atomic nuclei moving at near lightspeed and at rest). Also the prediction that clocks run slower deeper in a gravity well has been verfied by observation/experiment.
As "Doc Brown" might say, you have to think fourth dimensionally to understand how we can possibly look back 12 billion years into the past.