Verb
sleep with one eye open (third-person singular simple present sleeps with one eye open, present participle sleeping with one eye open, simple past and past participle slept with one eye open)
(figuratively) To remain vigilant at all times, especially by remaining partially awake or sleeping only lightly so as to avoid being caught off guard.
Scientists say crocodiles sleep with one eye open. Source: Internet
“The researchers found that crocodiles were more inclined to sleep with one eye open when humans were present, and that the open eye was always directed towards the humans,” reads a report detailing the team's investigation. Source: Internet