It is possible for a Complete Monster to also be an abusive parent; indeed I'd find it difficult to believe that any Complete Monster could be a good parent, but being an abusive parent in no way says anything about whether or not one is a Complete Monster.
It depends on the extreme of the abusiveness, and on what their other evil deeds are. I know Ozai's CM status as described on the Western Animation CM list mentions other things than just his abusiveness, and I know that for Frollo, his emotional abuse to Quasimodo (not his real son, but his adopted one) counts towards his CM status without said status depending on it.
Not to mention that abuse, as popularly understood these days, often has a Freudian Excuse behind it (namely, the cycle of abuse being carried on from previous generations), which is one of the Complete Monster disqualifications.
I think we've got a few more examples over here.
edited 2nd Dec '10 7:19:07 PM by feotakahari
That's Feo . . . He's a disgusting, mysoginistic, paedophilic asshat who moonlights as a shitty writer—Something Awfuli can't remember much from Moral Orel, but i think ozai counts. he does decide to engage zuko in an agni kai for something that wasn't even wrong in the first place (of course, since he's evil so he may see it differently), and burned and disfigured his son's face even though the boy had apologized and begged him not to hurt him.
of course, he's not a Complete Monster just because of that. there's plenty more in his repertoire. but it does add up. each case should be examined individually.
re: Freudian Excuse— it only invalidates Complete Monster status if it's an adequate justification. for example, i'm pretty sure ozai's father instilled some kind of complex in him; azulon did basically order him to kill zuko in cold blood, and he seems to have preferred iroh to ozai, but does that justify ozai's horrible actions later on? no. zuko himself proves that being abused and being The Un-Favourite are not enough to make one turn out a monster.
2 examples:
Are there any Abusive Parents whose abusiveness to their children goes to the extreme?