I don't really think that Gilgamesh is suspicious because of the way he talks to be honest. Because he didn't always use ellipsis and nobody mimicked his manner of speech on day 2 (i did on day 1 though). So to think he's a serial killer just because of that is not enough. He could be a vigilante or perhaps we're completely wrong about this.
I find him suspicious only because he was on Barthandelus's nice side.