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I don't think Izaya really qualifies as a complete monster.
Going through the criteria:
1. Personally commits actions- not really, most of the time he uses a bunch of people, both good and bad as pawns. While he's apathetic to the possibility of innocent people being harmed, he doesn't really do the harming himself (and has no problem with people escaping harm).
2. Terribleness played seriously at all times- Not at all. He's often presented humorously and scenes of him being villainous are rarer and therefore shocking. Also, while Shizuou (obviously) has murderous hatred of him, other good characters like Celty and Shinra are quite willing to interact with him.
3. No adequate justification- check for this one.
4. Devoid of altruistic qualities- Not sure about this one. I kind of got the feeling that while Izaya doesn't care much about whether innocents are harmed, he does enjoy opportunities to hurt bad people. I'd go 50/50 on this one.
5. No chance at redemption- IMO, the anime suggests he's fairly redeemable and that his actions are relatively harmless. After Simon punches him and gives him a stern warning, Izaya appears willing to moderate his behavior.
Not to mention his Pet the Dog moments in the light novel, where he also gets pretty close to having a breakdown at the thought of betraying his best friend, Shinra. Izaya has a Morality Pet, which almost outright disqualifies him as a Complete Monster. Furthermore, he does have a bigger reason for some of the stuff he does - it's all out of fear of The Cessation Of Existance. Though it reeks of Insane Troll Logic, it still counts as an excuse of sorts.
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How well does it match the trope?