Yes, a time-dilated object's clock can appear to move more slowly as a consequence of redshift. Redshift refers to the shifting of light towards longer wavelengths (lower frequencies) due to the relative motion of the source and the observer. This phenomenon occurs in both special and general relativity.
In special relativity, as an object approaches the speed of light, its motion causes a Doppler shift in the frequency of the light it emits or reflects. If the object is moving away from the observer, the light waves are stretched, resulting in a redshift. This redshift can be observed as a decrease in the frequency (increase in wavelength) of the light.
When an object is moving at high velocities relative to an observer, time dilation occurs, and the clock of the moving object appears to tick more slowly from the perspective of the observer. As a consequence, the frequency of the light waves emitted by the moving object is also affected, leading to a redshift. This redshift is a result of the combined effects of time dilation and the Doppler shift due to relative motion.
Similarly, in general relativity, the presence of a gravitational field can cause time dilation. Clocks in stronger gravitational fields run slower compared to clocks in weaker fields. When light travels from a region of strong gravity to a region of weaker gravity, it loses energy and appears redshifted.
Therefore, the time dilation of an object's clock can lead to a redshift in the frequency of the light it emits, resulting in the appearance of slower movement of the clock from the perspective of an observer.