Ariel as Aries, the archetype of anger
Aries is the Greek god of war, and in Greek literature, his name is frequently invoked in issues of divine wrath or true injustice. For Prospero, Ariel represents his connection to this archetypal force. He wants to do true warfare against those who betrayed him, and since his feeling is that he was truly betrayed, his anger is truly formidable, enough to literally invoke a "Tempest."
Alchemy and witchcraft
The thing about Prospero's Books is that they are clearly religious and alchemical texts. He has come a long way from Italian royalty. The books involve the sublime forces of nature and the metaphysical order of life. These things form a motif that codifies Prospero's willingness to experiment in magic, which is a demonstration to the reader of his absolute seriousness about his grudge. He has learned how to do real witchcraft, so he can one day have his vengeance.
The four elements
There are four individual characters who, together comprise Ariel, the sprite of Prospero's wizardry. These are three acrobats and a singer, all young, but of different ages. Their implied immortal nature is paired with adolescence because they represent elemental forces. The elements of nature are water, earth, air, and fire, and as an alchemist, Prospero's relationship to those forces is more than external—he is also governed by the same elemental forces. Perhaps Ariel represents an archetypal part of Prospero's psychology.
Caliban
Caliban represents something Prospero doesn't want to talk about. Caliban is differently abled, socially speaking, and there's that whole side-plot with Trinculo and Stephano portraying exactly that, so Caliban represents Prospero's difficult relating to people. He represents that Prospero is often made to feel like a monster, and he feels that people let appearances dictate their treatment of him. In other words, he struggles with feelings of disgust toward himself because he was rejected by people whose approval mattered to him.
The allusion to authorship
This film adaptation of Shakespeare's play The Tempest depicts an allusion to the author's artistic intent. The story suggests through the scene where Prospero furiously scribbles an epiphany, that perhaps Shakespeare wrote this play as a depiction of his own personal struggle to relate with community, given his genius, his closeness to his own nature, and his "witchcraft," which is language and art.
