Step 1: Identify the verb phrase.
The sentence uses the expression
\[
\boxed{\text{dealt with}}
\]
which is a fixed verb-preposition combination.
Step 2: Recall the usage.
The phrase deal with means
• to handle,
• to manage, or
• to resolve a problem or situation.
Since the complaint was handled promptly, the correct preposition is with.
Step 3: Analyse the options.
• (A) about -- Incorrect. "Deal about" is not a correct expression.
• (B) on -- Incorrect. It does not form the required phrasal verb.
• (C) with -- Correct. "Deal with" means to handle or resolve.
• (D) over -- Incorrect. It does not fit the context.
Step 4: Final conclusion.
\[
\boxed{\text{The complaint was dealt with promptly.}}
\]
Hence, the correct option is \(\boxed{(C)}\).