Chasing another person you know in a dream may be a sign of needing to protect that person from themselves. Many times people embark on paths that seem illogical or even self-destructive. The impulse is to chase them down and try to redirect them.
Another common chasing dream revolves around heroic behavior. You are chasing someone to right a wrong committed against you or another. You may not directly know whom you are chasing, but you know their relationship to someone else you love and value-be it their boss, a criminal who has victimized them, or some other antagonist. If you can't remember why you are chasing someone, try to concentrate on the emotion that the chase aroused.