You are absolutely right!
I found this entry on e Learn English Language, Online language learning with Laura K. Lawless:
Complement
Complement is a noun and verb. It also has a matching adjective,
complementary. In any case, it refers to something that completes or goes well with something.
The sauce is a nice complement to the vegetables.
The sauce complements the vegetables nicely.
This book will complement his collection.
I'm looking for a special shelf complementary to my book collection.
Compliment
Compliment is also a noun and a verb. It indicates the offering of praise or flattery to another person, as does the adjective
complimentary.
He gave me a compliment about my dress.
He was very complimentary about my dress.
Thanks for complimenting me.
Complimentary can also mean free.
The champagne was complimentary.
He gave me a complimentary copy of his new book.
The Bottom Line
The word pairs complement/compliment and complementary/complimentary are pronounced identically, hence the confusion in spelling.
Just remember that complement with an E means "to complete" or "to enhance," while compliment with an I means "to praise" and complimentary means "free."
HTH!