The phrase "chasing after" means pursuing someone or something, often with great effort or tenacity. Synonyms for this phrase include: pursuing, hunting, following, tracking, running after, seeking, searching for, striving for, and going after. Each of these synonyms expresses a similar desire to obtain something, whether it be a physical object or a person's attention. The intensity of the pursuit can also vary depending on the synonym used. For example, "hunting" typically implies a more aggressive approach, while "seeking" suggests a more passive, yet persistent, effort. Ultimately, the choice of synonym will depend on the context in which it is used and the tone the writer or speaker wishes to convey.