Ducks do that while sleeping. If you observe ducks sleeping near a pond you'll typically see them sleeping with one eye open and one shut. That way they can still stay alert enough to watch for predators.

I've had several pet birds and I've observed the same thing from them. I just assumed they were just pretending to sleep.

Your comment made me wonder how common it is with birds and it seems to be trait shared with all of them[1]. Pretty interesting. I asked my friend if her bird does that and it only seems to when it's not in a deep sleep yet, but eventually it will close both eyes.

[1] https://en.wikipedia.org/wiki/Unihemispheric_slow-wave_sleep

Gives a new meaning to the term, "sleep with one eye open at night." :)

Let me venture a guess: that expression comes exactly from observing animals keeping one eye open.

Haven't bothered to search for this, but do you know offhand if they also have a L-R hemisphere/body crossover?

All vertebrates have that, IIRC.