"Chased after" is a commonly used phrase to describe the action of running after someone or something. However, there are several other words and phrases that can be used as synonyms to convey the same meaning. "Pursued," "hunted," "trailed," "followed," "tracked," "dogged," "stalked," and "hounded" are a few examples. Each of these words implies a sense of urgency and determination to catch or capture the target. Some of these words may have negative connotations, suggesting a hostile or aggressive pursuit, while others may suggest a more positive or playful chase. Overall, there are plenty of alternative words and phrases to use in place of "chased after" that can add variety and depth to your writing.