To dream that you are being chased indicates that you are steering clear of a circumstance that you feel cannot be overcome. This is generally a metaphor for a kind of insecurity. If you dream that you are being chased by an animal, your anger that you haven’t displayed or acknowledged is placed within this animal. You could also be trying to escape some primitive urge or concern.
To dream that you are chasing someone indicates that you are struggling to reach a challenging goal or aspiration. You could also be showing belligerence toward others.