sleep with one eye open

Etymology
Some animals sleep with one eye open, so as to remain alert to threats.

Verb

 * 1)  To remain vigilant at all times, especially by remaining partially awake or sleeping only lightly so as to avoid being caught off guard.

Translations

 * Finnish: nukkua toinen silmä auki
 * French:
 * Polish: spać z jednym okiem otwartym
 * Portuguese: dormir com um olho aberto