應該用begin, 因為較配合全句意思 (两個不能同時進行的動作).
1. Have you ever seen the moon beginning to darken?
2. Have you ever seen the moon slowly turning red?
Analysis for the phenomenon of a lunar eclipse (total):
1. the moon starts to darken (when it is passing through the Earth's shadow)
2. then it changes its colour into red (when it crosses entirely into the Earth's umbra 本影)
3. the moon starts to brighten again
3. the whole process could take several hours to complete
The question -
Have you ever seen the moon slowly begin to darken and turn a red color for a few hours?
is more or less equivalent to asking
Have you ever seen the whole process of a total lunar eclipse?
What we know -
1. after the verb of perception such as "see", we can use either the bare-infinitive or present participle to follow the object, but they have different meaning;
2. inf. means witnessing the whole event;
3, pp. means witnessing the action in progress (that is, the action in progress is intercepted by the action of "see")