Example 1
I hate giving Jade compliments because instead of thanking me she always says, "I know, right?"
A compliment is one of those remarks that makes you feel totally good about yourself, like if you tell us that we make difficult grammar rules crystal clear (aw, shucks). So compliments is definitely the way to go here. (It's a safe bet that nobody's going to compliment Jade on her self-awareness, though.)
Example 2
"Aunt Edna, thank you for the scarf you sent me for my birthday!" Justin yelled into the phone. "It really complements the wool socks you gave me last year. Even though I lived in Florida, I wear both of them daily."
Since Justin, the model grandson, means that the scarf pairs well with the wool socks that Aunt Edna gave him, complements is the right word in this example. Now go toss that scarf in the same place you threw those socks, Justin.
Example 3
I used to think that Zack and Hannah made an odd couple, but now I can see that they totally complement each other.
In this sentence, the speaker isn't saying that Zack and Hannah say nice things to one another, like "Zack, I love your mullet!" or "Hannah, you make the best bologna and jam sandwiches ever!" The speaker is saying they make a great pair, so the e has it in this example.