What is an interesting character for you?
For me, what makes Mihawk interesting is his simplicity, he's a fighter, just a fighter. He is not mysterious like Shanks who is always involved with the most important men in the world, one hour he is talking to Newgate and another to Gorosei.
Mihawk is a fighter, a frustrated fighter. He is a man who found the thrill of living in fighting. He's like someone who plays extreme sports and gets addicted to the adrenaline rush of getting out alive.
Then he got so strong that he didn't find any more fights worthy of making him feel the thrill of being alive.
Shanks appears and becomes his rival and brings that feeling back to him.
But then, saving Luffy, Shanks puts Mihawk back in the harrowing situation of no longer being able to feel the thrill of fighting with all his might. Shanks gave Mihawk hope but was later responsible for taking it away, so we clearly see how Mihawk treats him with contempt for feeling betrayed. He mocks Shanks for not having an arm, in Marineford Mihawk doesn't even look at the man who was once his rival.
Mihawk is going through something similar to Kaido who hasn't fought at all since Oden, but I believe that for MIhawk it's worse because fighting is everything for him. He doesn't aspire to other things like Kaido. That's why his Vivre Card tells us he's looking forward to someone who will surpass Shanks, he wants to be able to fight for real and when he's close to death, feel how good it feels to be alive.
Anyway, I think I stretched too much, but this is Mihawk, I think if you expect anything other than strength you will be frustrated. I even think that bringing him anything beyond that would spoil the character. He is a fighter, a truly amazing swordsman according to the Vivre Card, bringing more information about his life just seems to me that the core of the characters was not well done. When you complete something more and more, it just seems to me that it was poorly designed. And I think Mihawk's strength was masterfully crafted.
Have a good day.