How is INDEX on Composite Primary Key in mysql?
The composite primary index will always ensure that the combination of col1, col2, col3 values are unique. If the purpose of the index is to ensure that the values are unique then you will have achieved that
A composite index also provides for index matches on any combination of the columns col1, col2, and col3 
You would create separate indexes on col2, and col3, if you join or filter using any of the columns without the others.
I always prefer a numeric primary key (with no business association), and unique indexes over a composite primary key where necessary.