Pursuit dreams are a fairly common type of anxiety dream. Consider what is pursuing you for clues as to whether it relates to alien aspects of yourself, primary or sexual urges, or perhaps to an inner battle with societal values. Although quite frightening, from Jungian and Gestalt points of view, the pursuer may actually be an unconscious or rejected aspect of yourself that is trying to make contact with, or be recognized by, waking consciousness, and may become more frightening if you resist its efforts. To recognize and attend to it and its purpose.
If you are pursuing someone or something, it may refer to your goals or ambitions; to pursuing the expression of elusive emotions; inner action or development that is taking you forward into the future; or if pursuing an animal to the freer acknowledgement or expression of instinctual urges.