The most common and most likely answer to this crossword clue is the 6 letter word RAILED
I've seen this crossword clue in Financial Times
R
A
I
L
E
D
More clues leading to the result RAILED
Buddy explains!

The straight part
We know that "had a complaint" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case had a complaint = railed.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does RAILED mean?
RAILED verb- To travel by railway.
- To enclose with rails or a railing.
- To range in a line.
- To criticize severely.
- To complain violently (against, about).
- (of a liquid) To gush, flow.
- Furnished with a rail.