Mihawk's only narrative purpose is to lose to Zoro. But because it's "simple" many people don't take it seriously. He's the guy who became the strongest at 19, a feat that will only be surpassed by the MC. He's the guy who gets bored fighting Shanks. Mihakw in a simple way is power, so much power that he can no longer have a serious duel.
Maybe Oda will make him defeat an admiral to show his power to the unbelievers and open up a spot for Smoker.