From the games to the comics, every assassin has had that extent on their hoods symbolizing the eagle, even Connor who was more related to wolves than the eagle has had it. Just like in Assassins Creed 3 fans complained that the hood should always be up, all assassins should have it (yes, I know Aveline did not, she's an exception). I think that following the tradition, this should keep going, the beak of the hood is almost as symbolic as the hood itself, and whether or not there will be other outfits with or without the beak, the main and most iconic one should have it.