Origin: Latin suffix -ary
Complimentary has 3 different meanings across 1 category:
costing nothing
"complimentary tickets"
"free admission"
In the nature of a compliment.
"The waiter offered complimentary praise about our delicious meal, saying it was better than any restaurant in the city."
In plain English: Complimentary means given for free as a gift or favor.
"The complimentary dessert was delicious and came with our dinner at no extra charge."
Complimentary comes from the noun compliment with the suffix -ary. It originally meant something given as a free gift or expression of praise rather than being paid for.