@Dark-Abel i do think Mihawk's motivations / characteristics are kinda weird for that kind of backstory
The three main things about Mihawk are:
1 - He enjoys fighting strong people, he enjoys the challenge. He didn't want to get strong for the sake of wanting to be strong, like Zoro, but he wanted to be strong so he could enjoy even higher levels of combat. Zoro is anxious of being weak and is always desperate trying to get stronger because of that, he thinks shit will go badly if he is weak.
2 - He enjoys solitude and doesn't like being in relationships / friendships with other people, he has slowly started to turn this around after seeing Luffy in Marineford. Allowing Zoro and Perona to live with him and now with Cross Guild. He doesn't trust most people.
3 - He is very observant and capable of analyzing people, he seems really interested in other people, despite wanting to be away from them in a way.
These 3 main things about Mihawk kinda paint a picture of a possible backstory for him. For example i think he used to have some kind of friendship / relationship that ended badly in the past, maybe it was Shanks, or another character we don't know. And that's why he isolates himself now. I think his overanalyzing people might be a coping / defensive mechanism, to try and protect himself from being betrayed or something like that again.