General Synonyms:
* Autonomous
* Self-governing
* Self-reliant
* Self-sufficient
* Free
* Unattached
* Unconnected
* Unbound
Synonyms emphasizing freedom:
* Unfettered
* Unconstrained
* Unrestrained
* Liberated
Synonyms emphasizing self-reliance:
* Autonomous
* Self-supporting
* Self-made
* Self-directed
* Resourceful
Synonyms emphasizing lack of influence:
* Unbiased
* Impartial
* Neutral
* Objective
Synonyms emphasizing financial independence:
* Self-employed
* Freelancer
* Entrepreneur
* Business owner
Synonyms emphasizing freedom of thought:
* Original
* Creative
* Innovative
* Thoughtful
* Unconventional
Synonyms emphasizing political independence:
* Sovereign
* Self-determining
* Free state
* Nation-state
The best synonym for you will depend on the specific context in which you are using the word "independent."