The phrase 'compliment on' is correct and usable in written English.
It is used as an idiom meaning 'give someone a compliment about'. For example, "I'd like to compliment you on your hard work this week".
"It's a compliment on one hand, and on the other hand, it's a little bit insulting," he said in an interview this week.
He paid me a great compliment on one of my jobs in Humbug Magazine, which also contained tons and tons of crosshatch.
Give a compliment on one of their edits.
"It's a compliment on your character," she told me.
"I always get a compliment on my cards," she said.
"It is not a compliment; on the other hand, it is not an insult.
He remembers receiving a hug and a compliment on his performance.
Even a compliment on two antique lithographs depicting stuffy British barristers -- the enemy within?
She answers his compliment on her perfume by laughingly explaining that it is deodorant.
He accepts my compliment on his office graciously – "Big and glamorous, yes.
The association, which represents Middle Village and Maspeth, proudly highlights the mayor's compliment on its membership forms.
