I've been staying away due to the crazy powerscaling debates, but, I wanted to say this:
Mihawk is a smart pirate, and knows Buggy and Shanks are connected/Nakama. If Crocodile actually killed Buggy, Crocodile would die by Shank's hand.
Mihawk is aware of this, and saved Buggy from Crocodile to save Crocodile from Shanks (and also because Mihawk is a bro and doesn't want Shanks to be sad).
Look at Mihawk's face in the bottom right, he's clearly thinking about Buggy.
Crocodile later says Mihawk doesn't trust any one, but, we know that's BS based on his encounter with Shanks/letting Perona and Zoro live in his house for two years. I think Mihawk has something planned behind the scenes with Shanks as well. Shanks and Mihawk following Luffy and Zoro so closely is bound to bring the two together.