A complex character is a character that shows signs of change, a character that evolves as the plot unfolds. The opposite would be called a flat or static character, referring to a character that remains the same throughout the story.
The villains in William Shakespeare’s plays are complex. Unlike the earlier antiheroes of the revenge or morality plays that were popular in Elizabethan and Jacobean culture, Shakespearean criminals lack the simple clarity of absolute evil. In his play Hamlet, Claudius is a complex character who shows the perfect example of a “quintessential Shakespearean antagonist”.
William Shakespeare portrays Claudius as a devious, greedy, and conniving villain who commits many sins throughout the play. However, in Act 3 of the play Claudius appears to feel genuine guilt over his murder of King Hamlet. Hamlet sets up a play with a plot that echoes Claudius' murder of the king in order to ascertain whether Claudius is guilty or not, via Claudius' reaction. But Hamlet is shocked to realize that not only does it confirm Claudius' guilt but also reveals Claudius' genuine culpability over what he has done.
Claudius is socially adept, and his charm is genuine. He can exhibit deep distress over his "dear brother's death" and admiration for his wife, "Th'imperial jointress to this warlike state." He knows the value of a great funeral, but quickly turns mourning into celebration and moves on "With mirth in funeral and with dirge in marriage" to whatever lies ahead. He is a decisive man, fair in his politics and commanding — if Gertrude's allegiance is any indication — in his bedroom. This proves the statement that Claudius is a complex character.