Alexa: I think they both force the other to be honest with one another. They don't keep secrets from one another (usually) or lie to one another. They have no reason to. They are co-leaders that helped keep everyone safe in the beginning, they aren't in a relationship... so they have no need to keep secrets. And I think they ground each other. They have the other to bounce ideas off of and the other to reel them back in if things get out of hand.
Paula: She seems to be the only one who can really get through to him, make him see reason. He doesn't get to make rash decisions with her around. Which is better I think 
I always like Clarke more when she's around Bell 
I think he brings out the best in her because he supports her.
Kate: I am trying to not ONLY think of s03 here but it is almost impossible. In s01, their roles as head and heart were less defined I think (though it’s been a while for me so forgive me if I am completely off track here). I am thinking of the hang-Murphy scene. She was very much just emotion in that scene without realizing what mob mentality could bring about, whereas Bellamy (maybe because he is slightly older and more experienced) was far more cautious. She learned a valuable lesson in that scene and his screaming at her (“This is all on you princess” or sth to that effect) really hit a nerve with her imo. In general though, it is a give and take between them. She makes him see the good things about himself, gives him confidence and faith that he did not have. He reins her in when all she can see is the desired end result of a situation (no matter the cost), making her see that the way you get there is pretty much equally important (I am thinking of her desperate plan to find someone to give the flame to, and he is like NO, we’re NOT doing that). All in all they complement and complete each other (eugh how cheesy does THAT sound
)
Adie: they both ground each other. trey trust each other.
Ana: She makes him believe in himself, and trust he can be that person he wants to be, instead of the trainwreck he believes he is. Bellamy keeps her grounded in a way. She's all head, and he gives her human warmth. He keeps her human and aware of her own feelings, weaknesses or not. Also, he is the one person who calls her out on her wrongs and that she will listen. He is the one who makes her understand what she was doing wasn't right.
Sarah: Clarke brings out the best in Bellamy by simply believing in him, something that no one else really has done and in doing so, he starts believing in himself - that he's a good person who has messed up many times but ultimately means well and simply wants to protect people he loves/cares about. And we all see what that unspoken faith that Clarke has in him has brought out in him, he was the main reason why the MW rescue was able to take place, before Lexa upped and betrayed them of course.
It was because of his own abilities of course and also Clarke telling him that she knew he could pull it off. "You came through, I knew you would."
As for how Bellamy brings out the best in Clarke, he makes her a more compassionate/open person. She is considered a person who's quite closed off emotionally (due to her past experiences) but around him, she just feel comfortable enough to open up and be vulnerable and honestly that is about the most intimate thing you can do with a person apart from the physical affection (which they have anyway)/kissing/sex part.
Lisa: Clarke brings out the best in Bellamy by letting him know he has worth, plain and simple. That alone gives him strength and makes him want to be a better person. All the things he's done to help their people has been partly due to her encouragement. Clarke has had the most scenes with him talking about thoughts feelings, etc.
Bellamy brings out the best in Clarke by teaching her that she doesn't have to do everything alone, that there are people there for her always. He would back her up always, would never betray her and never lie to her. I think in a way he helped her begin to trust people again.
Lee: Clarke brings out the best in Bellamy because when they are around each other Bellamy has the second voice to tell him the facts behind the situation rather than just what he was feeling in the moment. He has always been a person to go with his gut and with Clarke she has always looked at the facts behind it. By themselves is not a good situation as we saw in 3A. They make their best when they can see all sides of it. Which is only something that he can do when they are together. Bellamy also helps Clarke look at the people rather than the war. Clarke's thoughts are merely statistical the statistics do not always have the best outcome when the people you care about in trouble.