I think that "belief" and "morality" should the core of a complete monster. Not to say that s/he follows the same beliefs and ideas that we do, no no no! A complete monster has their own ideas of belief and morality, much like many vampires of the Sabbat had did in Vampire: The Masquerade, which could range from dipping yourself into as much depravity as possible to becoming a saintly type of vampire -- a twisted take of what it means to be a saint, granted, but a saintly path you can follow.
What your monster believes in -- a cause, a notion, etc. -- gives him direction, where to proceed from here. The Operator from Serenity, though not listed as a Complete Monster by TV Tropes standards, refers to himself as a monster who would not be accepted into The Corporation's utopian paradise, but he believes in it so strongly that he will murder and destroy anyone that stands against that dream. In fact, it's because of the wiliness of Mal and his crew that he orders a Scorched Earth policy against them, and has every place they have ever taken shelter destroyed, every person they had connections with murdered, all so that they would have nowhere to run and nowhere to hide. It is as horrific as it sounds, especially when they find that Shepard Book's haven has been wiped out, and everyone, including children, have been executed. The Operative's reason -- "You left me no choice." Not even a slightest bit of remorse because, to him, it's all just another part of the job.
If Belief is your complete monster's compass, then morality is how s/he applies his belief. The Joker, from The Killing Joke seeks to break Gotham PD Commissioner Jim Gordon to prove his thesis to Batman that anyone can be pushed to the depths of madness and become just like the Joker from One. Bad. Day. Because of his belief, he first goes to Gordon's house when he and his daughter are home, rings the doorbell, then shoot's Barbara through her spine, capture Gordon, then sets him up for the main event -- Gordon strapped to a Rollercoaster ride naked while being shown the various Polaroids Joker took of the crippled, bleeding Barbara before and after he strips her naked, all while taunted the helpless Commissioner. It's absolutely maddening, as Gordon has no idea if his daughter is even alive, and then faced with the possibility that Joker molested her body... or corpse. Wouldn't you beat Joker to death with your bare hands the moment you got free? Well, that's the whole point -- to break the Commissioner, strip away his self-righteousness as a police officer and a servant of the Law by robbing him and his family of dignity. mock his weakness, rile him up to the very brink of sanity, and give him that one tiny push over the edge by getting him to murder Joker in cold blood. Even if Gotham City forgave him unconditionally, there's no way Gordan will be able to live with himself, and thus, he'll destroy himself if not become like the Joker.
Overall, a complete monster isn't just a psychopath or an anarchist -- he's so much more. It should be terrifying if he smiles at you and offers you a nice hot bowl of chicken soup while you're worried about the girlfriend that hasn't been seen in the last week. Chances are, he's waving the answers you crave under your very nose, and you won't see it until you look twice.