There are several synonyms for the phrase "chases after" that can be used to convey the same idea in different terms. Some of the common alternatives to this phrase include pursues, follows, trails, tracks, hunts, seeks, and runs after. These synonyms can be used interchangeably to describe someone or something that is actively and persistently trying to catch up with, capture, or attain a desired object, person, or goal. Regardless of the specific term used, the concept of chasing after something or someone implies a sense of determination, passion, and energetic pursuit that is characteristic of many human endeavors and pursuits.