Sentences with the verb “look” (in the sense of “appear, seem”) have a different structure from what many English learners think. “Look” is followed by an adjective, not an adverb:
However, we run into another problem in the case of “good” and “well”. “Well” also has a second meaning, “in good health”, which is used as an adjective. Therefore, both sentences below are grammatically correct:
The meanings are as follows:
he looks well = he seems to be in good health
If we use “look” in the sense of seeing or searching, then “look well”, where “well” is an adverb, makes sense. For example, we could say:
This construction is possible, but “carefully” is more common in this context: