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Guile Hero: To some extent. While he's not The Chessmaster, during Caligula's reign his Obfuscating Stupidity alone wouldn't be enough to save him, thanks to Caligula's cask-strength insanity and psychotic urges. Several times he saves his own life, as well as the lives of others around him, by manipulating Caligula with not-inconsiderable skill. He's also a skilled administrator, and manages a long and (mostly) successful reign as Emperor largely due to extreme competence in the face of being loved by almost nobody.
Historical Hero Upgrade: Claudius and his desire to restore the Roman Republic, which is also somewhat unhistorical, since at that time there was no distinction made between the Republican and Imperial eras.
Lonely at the Top: The isolation involved with being a member of such an important family is one of his major problems. Becoming Emperor certainly doesn't help it, either.
Not So Above It All: A darker version of this trope: Claudius thinks he can remain separate from the murderous schemes absorbing his family. Unfortunately, when Claudius himself comes to power, he finds he must get his own hands dirty in order to survive.
I Did What I Had to Do: Livia ruthlessly manipulates and kills family members and anyone else close to them to ensure her son becomes emperor and Rome does not return to being a Republic, convinced this is the only way for the city to remain great.
The Extremist Was Right: Cladius has a great deal of personal hatred for Livia, but Deifies her anyway after being Emperor and learning how much work she did to make everything work.
Ignored Epiphany: Admits she was a terrible criminal and person, but decides it was all worth it in the end.
I Was Quite a Looker: Livilla mentions that she was said to be the most beautiful woman in the world.
Hypocrite: When things start going seriously down the drain she starts criticising everyone around her for not having the courage to kill Caligula. At no point does she even contemplate the notion that she might give it a try.
Offing the Offspring: Locks Livilla in her bedroom to starve to death after learning of her role in Sejanus' plot.
Parental Favouritism: Germanicus is obviously her favourite. She doesn't even seem to like Livilla all that much, even before Livilla becomes her grandmother's pet.
Soapbox Sadie: Is constantly bemoaning how Rome is going down the toilet.
Hair of Gold, Heart of Gold: A rare male example of this trope; blond-haired and admired by all as a virtuous hero of the German front. His children are all inversions (either good but dark-haired or blonde but evil).
Hair-Trigger Temper: After going insane. Celebrate the anniversary of the battle of Actium, and he throws a fit. Don't celebrate it, and he throws a fit. Please don't mention the fact that his grandfather was a commoner.
I'm a Humanitarian: He cuts out and eats Drusilla's unborn child, fearing it will become more powerful than he.
Insane Equals Violent: Subverted. His violent/psychopathic tendencies are explicitly shown NOT to follow from his psychotic delusions: he's a killer from childhood, but doesn't go mad until after he becomes Emperor years later. Livia and other murderous characters are described as "mad" by other characters, but are not shown as irrational - even Nero, explicitly called "as mad as... Caligula", is clearly nothing of the kind.
It Amused Me: Caligula has some shades of this - he does things like set up the young, beautiful Messalina with unattractive Claudius because he thinks it's funny.
Spanner in the Works: In more ways than one. Livia consents to him being made Tiberius' heir because of a complex scheme that will result in her elevation to godhood (and thus avoid eternal damnation). When Livia is on her deathbed, Caligula gleefully turns on her, and later comes very close to destroying the empire Livia worked so hard to build.
Sergeant Rock: "They say your drills are bloodless battles, and your battles are bloody drills." Tiberius, however, is fair, even sleeping without a tent if his troops couldn't have one.
Sketchy Successor: Played with. As a heir to Augustus he is for the most part portrayed as a depraved tyrant; yet Claudius acknowledges that he was competent at governing the empire and that the majority of the population had little reason to complain during his reign, with only the minority suffering from his repressions. However, the trope is played completely straight during the last years of his reign, when he just stops caring about the administration of the empire whatsoever.