There are several synonyms for the phrase "keeping busy." One of the most common alternatives is "staying occupied," which implies that a person is engaged in various tasks or activities to avoid boredom or idleness. Another synonym is "chasing distractions," which suggests that someone is focusing on different things to keep their mind off of negative thoughts or feelings. "Working hard" is another synonym, indicating intense focus and determination to achieve one's goals. "Dabbling" is another synonym that suggests experimenting with various activities or hobbies without a specific goal in mind. Finally, "filling one's time" implies that a person is simply finding different things to do to pass the time.