What does it mean to dream of chasing a Basset Hound?

There are as many different opinions for interpretation of dreams as there are people that do it. It depends on who you ask.

The mechanism of why we dream is not fully understood - by anyone. That it is a function of the sub-conscious is fairly well agreed upon but there are many theories as to why and how we dream.

People have different solutions with the conscious mind when presented with the same data, it is far more difficult to explain the functioning of the unconscious one.

It is, however, a practical certainty that there is no pre-cognition involved.

A different response:

This dream might be a play on Words, related to the expression "being dogged by a problem." There is some issue in real life that just won't go away - it keeps "dogging" you, following on your heels like a hound that won't give up. In the dream, stop running, turn around and face the dog. In real life, the solution is to figure out what the issue is, then stop avoiding or running away from it.