The bird that can sleep with one eye open are Mallard birds.
Mallard birds are birds that sleep in groups and often sleep with one eye open if they're on the outside of the group.
The open eye of the Mallard bird is directed away from the other birds.
Other birds that can sleep with one eye open are Geese, chickens and Oystercatchers.
Oystercatcher birds have an eye blinking strategy that is called "peeking" which allows the bird to sleep while also periodically opening one of their eyes to monitor their surroundings.
Chickens also often sleep with one eye open.
Oystercatcher birds are a group of waders that form the family Haematopodidae, which have a single genus, Haematopus.
Oystercatcher birds are found on coasts worldwide apart from the polar regions and some tropical regions of Africa and South East Asia.
The oystercatcher bird is a boldly patterned shorebird that has red-yellow eyes and a vivid red-orange bill, American Oystercatchers survive almost exclusively on shellfish—clams, oysters, and other saltwater molluscs.
Because of the oystercatchers specialized diet, oystercatchers live only in a narrow ecological zone of saltmarshes and barrier beaches.