Being chased in a dream usually points to something you're avoiding rather than an outside threat -- an unresolved conflict, an emotion, or a truth that keeps catching up no matter how fast you move. Turning to face whatever's chasing you, even in the dream, often marks a readiness to deal with it directly.
Destiny is not a matter of chance, it is a matter of choice; it is not a thing to be waited for, it is a thing to be achieved. (Bryan, William Jennings, 1860-1925, American Lawyer Politician)