Yes, he sleeps this way. I swear to you he's not dead.
He's a Fighting Fish/Betta. His name was originally Pebbles, then Rainbow, but now it's Sleepy. He will forever be Sleepy now. I guess he's sleeping anyway. He's definitely not dying. I thought he was dead the first time he did this. I'm not sure if this is a normal thing, but I have never seen a fish sleep like this in my entire life. My grandmother either. It's freaking hilarious. I crack up every time I see his butt/fins sticking out of his little house. It's so funny.
We got him a few weeks ago though, so hopefully he lives a long life unlike the previous fish in that bowl. I also have a betta fish named Blue who's been around, but he's not as interesting as Mr. Sleepy. :P