Being chased means the dreamer is trying to escape a part of their own unconscious personality...

21 Days of Dreams

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In a dream, the unconscious mind of the dreamer is projected outside, creating the figures and symbols of the dream world.

A person or creature chasing the person in a dream is in most cases a part of the dreamer’s personality that they need to accept. 

Another possibility is that the creature doing the chasing is an instinct the person is denying. 

For example one time I dreamed of a lion chasing me, so I climbed a tree to try and escape it.

The dreamer’s resistance and rebellion against the part of themselves that they encounter sets them in motion to run away, but the unconscious mind will chase the dreamer and attempt to complete it’s process of compensation – balancing the dreamer’s attitude by supplying the ‘other-side’.

On reflection, the dreamer when they wake up should try to learn the qualities of their pursuer and learn that it is a part of themselves that has a part to play in their life.

The lion represented the inner instinct to defend myself against death. 

It turned on me in the dream and chased me because I was denying its existence and the role it needed to play in my life at that time.

 My financial situation was very bad and I was in a foreign country where no one could help me.

My normal attitude was that things were okay, but the dream indicated that this instinct needed to be accepted and I should pay attention to getting security back in my life.