A complex personality is simply one that features many facets or levels. A personality complex, according to the renowned psychologist Karl Jung, is a fixation around a set of ideas.
According to a scholarly article written by Mihaly Csikszentmihalyi and published by Harper Collins, complex personalities are often associated with creativity. Creative people adapt well to their surroundings. They are able to reach their goals regardless of adversity. Creative people often show extremes in their personality, for instance being both learned and naive or both playful and disciplined. Such complexity of personality allows for them to think differently than others and to be creative.
Complex personalities can also lead to idealism. In this case, they are often in touch both with their own thoughts and feelings and with human nature.
Personality complexes are the other side of the coin. Some people develop a set of emotionally charged ideas that reside in the subconscious. Such complexes often function as a subset to the dominant personality and influence a person's behavior. Complexes can be negative or positive, but they are often associated with negative behaviors.
Complex personality disorders occur when the underlying complexes take over and negatively influence a person. For instance, a superiority complex might influence a person to express behaviors that seem conceited.