To me there is no ridiculous question.
Light can behave as a wave one time and a particle other time depending upon how you probe it. If you observe its interaction with electrons, it would behave like a particle (photon). If you allow coherent light source to pass through multiple narrow slits placed close together, it would behave like a wave (interference patterns). So to probe light you have to interact with it. A photon does not have internal structure and hence no mass. When it moves from one point in space to another, it does in no time. So essentially a photon takes all possible paths. From a photon's perspective time does not flow. But the space and time are not independent entities. they are connected by a metric that limits the speed at which 2 events can be causally connected. Light travels at that cosmic speed.
So to answer your question, the metric of space and time does not change for inertial observers and so you will see the light speeding away at the cosmic speed no matter how fast you run along with it. This also means that light moves at an aboslute speed! But this creates paradoxes... which is resolved by the special theory of relativity