Researchers from La Trobe University and Max Planck Institute for Ornithology recorded young crocodiles and found that while the animals typically sleep with both eyes closed, in the morning hours they will sometimes open one eye periodically. If they placed another crocodile in the tank or stood nearby however, the animal would open one eye and look at the newcomer.
It was noted in the article that certain birds and aquatic mammals will likewise sleep with one eye open, with only half their brain staying awake. This is known as unihemispheric sleep. This allows the animals to watch for predators while the other half of the brain sleeps.
Without measurements of brain activity, it is unclear if the young crocodiles also experience unihemispheric sleep or if they are simply keeping a eye on you.
Source:
"Without measurements of brain activity, it is unclear if the young crocodiles also experience unihemispheric sleep or if they are simply keeping a eye on you."
Sounds like you have the next graduate student challenge well in hand. See that croc? Yes ... the one on the mud bank ... the first one to attach leads and get a good reading gets the weekend off.