I've always seen depth of character, in this case a villain, as to be one of three things;
1: Explicit narration - The worst kind of deep.
e.g. "They want the thing because [insert reason here]..."
2: Allusion through protagonist agency - The best kind of deep.
e.g. She went to the apartments of where the suspect stayed over the last twelve months. Little nooks and crannies of the landscape, mostly dilapidated, and all with views of a city that one could only remark as monstrous. Encompassing its inhabitants within in its eternal hunger for money, land and people. Producing only iniquity and unbridled decay in its rampant consumption.
She didn't find anything concrete, but she felt as if she could get inside his head now. What he was thinking as he gazed out over a city he felt alienated from, that he would come to loathe, as he hatched that scheme of his in his head--or possibly even before it.
She was convinced she had her man, but now she had to prove it.
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Basically presenting the above without directly telling someone...
3: A combination of 1&2 - The most difficult kind of deep, and can possibly be the best or worst.
e.g. "They still don't understand. Why can't they understand!?"
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I'm going to give an example out of anime, to prove that I'm fair by giving you an example of an anime despite not liking it, and show you what I mean of no. 3 of a really deep villain...
And that is Patlabor 2 ... but actually both Patlabor 1 & 2 are really good movies with really good villains. One of whom is already dead before the movie even really starts ... it's brilliant. It's like a forensic examination of the mind of a madman, trying to piece together what they were trying to say about the nature of the world, and from that determine the motives of their activity and what they did and hoping to achieve.
It really really works...
In Patlabor 2 the villain ofthe piece is introduced right at the start. A Colonel Tsuge as part of the Japanese Ground Self Defence Force detachment connected to a UN peacekeeping mission 'Somewhere in Southeast Asia' ...
The mission is an utter catastrophe, and Tsuge is assumed to be one of the few survivors of that particular skirmish.
I won't spoil the movie, but from there through the orchestration of a couple of terrorist attacks, exploiting the corruption within the various branches of the Japanese police and shadowy connections between the military and government that date back to the postwar period ... Tsuge essentially 'brings his war back home.'
He orchestrates a no-win scenario through his strategic use of terrorism, technology, and banking on the corruption at the heart of Japanese government, and essentially creates a no-win political crisis event. Whereby the government will likely collapse under its own weight of deception.
No one wants to back down, no one wants to surrender, and no one wants to shoot and invite a catastrophe. And so the collapse of civil government by default.
A situation he himself felt betrayed to by his superiors in Southest Asia that saw some of his men die.
And fr most of the movie the protagonists are merely reactive to this plot, asthey themselvesare embroiled between a political stand off by the Japanese Diet, military who feel unjustly villified and persecuted for incidents they had no hand in, and the police who out of that corruption acted rashly and tried to arrest key members of the JSDF.
One that eventually leads to the concession of the Japanese Diet, a vote of no-confidence in the police (of which nearly all of the protagonists belong to), and martial law. It's a pretty good ... I won't call it a pure 'political thriller', but it's in that genre nixed in with 'cop drama'. There's very little action beyond the first few minutes, and the brief bursts of orchestrated horror and the finale ... It's just a good villain in a pretty good movie.
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I will say that because of movies being a visual medium predominantly, that allusion is easier to create in videos than in books. You can present the scene from first person perspective as you navigate the ruins of a villain's childhood home without either the protagonist or a writer just narrating at you.
Through careful shot composition and good acting and direction, you can also create a dual visual narrative of a marriage between environment and the person that sells an indefinite connection between protagonist and antagonist as they feel an almost shared event beyond thestrictures of time and distance.
Which helps illustrate the differences between both character(s) in full.
With clever use of a soundtrack and muted background and speech, you can even almost construct an alien mindset in the viewer. Make them isolated and slightly uncomfortable in the process. Which is particularly difficult outside an audiovisual format.