The situation you described involves both time dilation and the Doppler effect, and the combined effect depends on the specific conditions of the scenario. Let's examine each effect separately and then consider their interplay.
Time Dilation: According to the theory of special relativity, as an object moves relative to an observer at a significant fraction of the speed of light, time appears to pass more slowly for that object relative to a stationary observer. This is known as time dilation. Therefore, a clock moving towards you at near light speed would indeed appear to run slow from your perspective.
Doppler Effect: The Doppler effect refers to the change in frequency or wavelength of a wave (including light or sound) as perceived by an observer when there is relative motion between the source of the wave and the observer. If a clock is moving towards you at a significant fraction of the speed of light, the light waves emitted by the clock will be compressed, resulting in a blue shift. This means the frequency of the light will increase, and the clock will appear to run faster.
Now, when considering both effects together, the situation becomes more complex. It's important to note that time dilation and the Doppler effect are not contradictory; they are both valid phenomena in relativistic physics. However, the specific outcome depends on the relative velocities involved and the frame of reference being considered.
If the clock is moving towards you at a relatively low speed (compared to the speed of light), the time dilation effect dominates, and the clock would still appear to run slower due to time dilation, despite any additional effects from the Doppler effect. In this case, the time dilation effect is stronger and overrides the Doppler effect.
However, as the relative speed between you and the clock approaches the speed of light, the Doppler effect becomes more pronounced, and its influence on the observed time becomes significant. At extremely high velocities, the combined effect of time dilation and the Doppler effect can lead to complex changes in how the clock appears to run, depending on the specific velocities involved.
In summary, the outcome of how a clock moving towards you at near light speed appears to run depends on the specific velocities involved and the interplay between time dilation and the Doppler effect.