We went to watch a movie after we had completed our work.
Why the need for the word "had" when there is the word "after"?
You would still have "had" if you used another term, even an opposite term, such as "before."
We went to watch a movie before we had completed our work.
"Had" is required as a part of the pluperfect construction. "After" is a preposition, like would be "before," that signifies the temporal relationship between the clauses. It adds semantic value and as demonstrated is generally independent of the tense.