Inhuman Emotion

A character with emotions or feelings that are beyond human (well, most humans), or, at the very least, different from humans. Their pain and pleasure can be magnified to points we would find unbearable, they might have an extreme sense of empathy, or they might even have emotions that don't even exist for us. Or all of the above, and more.

It's often found in a Superior Species, because what better way is there to denote superiority than simply being able to experience more than you. Likewise, an Eldritch Abomination almost always has this aspect as well, usually to shape part of their Blue and Orange Morality.

Characters with Inhuman Emotions are prone to having Super Senses as well.

Comics

 * Kind of the point of Supergod.
 * In the final issue of All-Star Superman, it turns out that Superman's empathy and altruism are also superhuman due to his enhanced senses. The temporarily-empowered human  finds this out personally and
 * According to Doc Samson in an issue of Thunderbolts, everyone with gamma powers has an inhuman level of rage within them. When a telepath tries to get into his mind, he overwhelms her with a dose of his.

Literature

 * The cherubim Proginoskes in A Wind in the Door by Madeleine L'Engle. (Yes, 'cheribum' should be plural).
 * In Foreigner, the social structure of the atevi is determined by the emotion "manchi", which is not exactly understood by humans, though it has similarities to human notions of loyalty.
 * All over the place in Clifford Simak's books with his numerous super advanced aliens that the human protagonist befriends.
 * In L. Jagi Lamplighter's Prospero Regained, the characters reach the deepest pits of Hell and feel the temptations of the creatures imprisoned there -- the insane temptations.

Live Action TV

 * A mild example in Star Trek, as Sarek (a Vulcan) says that "emotions run deeply within our race, perhaps more deeply than in humans."

Tabletop Games

 * The Eldar in Warhammer 40000 are forced to be The Stoic as a race, because they are so incredibly emotional, whenever they allow themselves an outburst their evil god rips their souls from them and devours them.
 * Space Marines also feel emotions more strongly than normal humans. (It's mentioned that the fan-created Angry Marines feel rage to an extent that boggles even other Space Marines.)

Video Games
"Forgiveness Occasionally, a captive dragon will forgive its captor and perform an act of kindness in a time of need. The dragon is in every way a superior beast to the human: superior in mind, superior in body, superior in heart. When a dragon shows anger, it burns brighter than any human rage, but draconic forgiveness shines even brighter. Only the most callous of captors could resist releasing such a noble beast."
 * In the flash game How To Raise A Dragon by Gregory Weir, the player plays a dragon. During the 'adolescence' phase, the wizard who captured the dragon as a hatchling is lying sick in bed, leaving the dragon free to choose a breath power and escape. However, if the dragon chooses the healing power, it can cure the wizard.

Web Comics

 * Cultural example - the trolls in Homestuck have a different perception of romance to humans, dividing it into four different kinds, with two of them sexual and two of them platonic; and consider true hate as erotic and stable an emotion as they consider true love. For the most part, the feelings themselves aren't that alien - what's alien is that trolls embark upon, say, negotiating between two people in a destructive romance with the same ferocity that we fall in love with people. In this case, it's Blue and Orange Morality, and the trolls are simply different from humans - not better, and not worse.
 * Goblins mentions that a demon can suffer more than any mortal, at least in one character's reasoning.

Western Animation

 * Raven's anger in Teen Titans. When she loses control and her demonic side takes over, she sprouts dark tentacles of black magic that can drag victims under her cloak, where they are psychologically scarred.
 * In one of the Futurama episodes with the what-if machine, Bender references the emotionless state of a robot: "Robots don't have emotions, and sometimes that makes me sad." In comparison to robots, at least, he may be exhibiting Inrobot Emotion.