The Complete Monster can sometimes be indicative of lazy writing: a villain with no redeeming qualities can be viewed as exceedingly simplistic. A particularly poorly-executed Complete Monster will fail to engage the audience.
The story's context is important in establishing such characters—what may seem like an act of ultimate evil in one story could be just business as usual in another. The reaction to such acts by other characters is usually most telling.
The tone of the work and its' characters is also to be considered. There is a big difference between a Laughably Evil villain who still commits heinous acts that are only perceived as funny by the villain and a villain whose character and actions are Played for Laughs by the narrative, which is shown by how little a reactions the actions actually get from the other characters. A Complete Monster can be both reprehensible and Laughably Evil but neverPlayed for Laughs.
The scale of a villain's actions, the playing field they occupy, and the resources in their possession are important to note. If the villain is an Evil Overlord with enough power and resources to cause widespread devastation and does so, then this character is a example of the trope. But if the villain is a low level Serial Killer who causes as much devastation within his/her setting as is possible given their limited power and resources, than this character is also an example of the trope, for it suggests that a smaller playing field and lack in resources is the only thing keeping this villain from doing worse, so they're as bad as they can possibly be with what they have at their disposal as compensation. Therefore, it is entirely possible for varying degrees of Complete Monster to exist within the same universe without truly eclipsing one another in heinousness. (Darkseid and The Joker are good examples of this.)
When applying the heinous standard of any given work to a candidate for this trope, a bar graph must be visualized. A villain with a rap sheet of deeds that are treated as the most heinous sort of evil by the work's story is the bar-setter. A villain with that AND no redeeming features to speak of is an example of the trope, thus becoming the work's heinous standard. If the work features more than one example, then those examples have to equal or even surpass the villain who's at the top of the bar in order to qualify.
Moral agency and the capacity to make choices is a requirement for this trope. Groups cannot qualify for the trope, only individuals with moral agency can. Neither can beings Made of Evil or animals qualify because they act upon instinct, not choice. A true Complete Monster has to be a villain who has all the capacity to not do evil in the world, yet consciously decides to be evil and commit the absolute worst kinds of evil for inexcusably selfish and petty reasons.
Not only must an example of this trope lack a Freudian Excuse that's adequate enough to explain away or justify their present day deeds, but they must also lack any truly redeeming qualities. If they had any such qualities in the past, they lack those specific qualities in the present.
A character crossing the Moral Event Horizon does not alone make them a Complete Monster. This trope isn't just about what the character does, but about what the character is. Their monstrous characters are reflected in their heinous deeds, which is what puts them a cut above the regular villains. And whatever their position, a Complete Monster has to go the full mile and meet all criteria: they are the worst they can possibly be in their role, in the space and scale they occupy.
While Offscreen Villainy is advised against, a few exceptions can be made depending on how clear the results of said villainy are. Gory Discretion Shot is common for this, as are explanations of the monster's deeds being coupled with displays of results that speak for themselves.