I’ve made a simple anglerfish, but I don’t know how to make more realistic deep water. Any idea?
Try putting some tiny floating particles, that are only lit by the light on the tip of the fish.
you could add some tiny bubbles
you could add some small fish in the background, defocus them a little, and add some graininess to them.
Maybe add some overall grain in the water to simulate little particles.
Search for some photo references and add the effects you like. You could try adding some procedural textures to the background (Voronoi, Musgrave, Stucci, etc.).
Oh man; that is awesome!
tadpoles, bubbles, currents with tiny particles of trash or bacteria, decaying organisms…etc…all of these things make water look like ocean water instead of bath water…try a particle system…hdri of a coral reef, model some shrimp, or tiny baby fish…
Have you tried blending your background between black and dark blue, adding some mist with quadratic scaling and with a max depth of 300 meters.
This will give you a simple deep water feel.
Good luck.
Another thing that is important, add a volumetric halo spotlight with a texture, so that you can see light rays coming from above.
There was a tutorial for underwater… I will see if I can find it! 
Got it! Use this maybe?
Keep it simple. At the moment it’s a cute simple character, so don’t try and add a million and one things, as you’ll only kill the charm of it. The only thing I would suggest is playing with the background colour, maybe tweaking the colour balance a little to get a little more blue for an underwater look.
Ya, i like it simple too. You are not going for realism here so dont try to make it look too realistic. Bottom line? Just like ben said, dont add to many things. keep it simple, at least thats my oppinion.