Fish do sleep with their eyes open, because they don't have eyelids (except for some sharks) to close! Fish sleep is not exactly like human sleep, though. For starters, they don't use pillows.

How can you tell if a fish is asleep?

It's pretty easy to tell when fish are sleeping: they lie motionless, often at the bottom or near the surface of the water. They are slow to respond to things going on around them, or may not respond at all (see some sleeping catfish here). If you watch their gills, you'll notice they're breathing very slowly.

Do fish float or sink when they sleep?

The nature of fish "sleep" is an area of active research. While fish do not sleep in the same way that land mammals sleep, most fish do rest. ... Some fish float in place, some wedge themselves into a secure spot in the mud or coral, and some even locate a suitable nest.