Green pulse on left hand every so often - why?

All of the Vigors have this sort of periodic reminder. Your hand will show feathers and long nails when you have Murder of Crows equipped, glow red and start to burn if you have Devil's Kiss equipped, etc.

There's not really any rhyme or reason as to when this occurs, it just does every so often. I like to think that the game is just reminding you that you've got Vigors and could use them if you wish.

It's a flavour animation to remind you of which Vigor you have equipped. For Posession it's a green pulse, but it will change for the other vigors.

Edit: the original Bioshock did the same thing with Plasmids.