The most common and most likely answer to this crossword clue is the 12 letter word TUBERCULOSIS
I've seen this crossword clue in Financial Times
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More clues leading to the result TUBERCULOSIS
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The straight part
We know that "Complaint" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case complaint = tuberculosis.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 TUBERCULOSIS mean?
TUBERCULOSIS noun- An infectious disease of humans and animals caused by a species of mycobacterium, usually Mycobacterium tuberculosis, mainly infecting the lungs where it causes tubercles characterized by the expectoration of mucus and sputum, fever, weight loss, and chest pain, and transmitted through inhalation or ingestion of bacteria.