Weird Aside

During what had previously been a perfectly ordinary conversation, one character nonchalantly inserts something completely bizarre, and often disturbing, which often derails the conversation. They may do this knowingly or innocently, not realizing how strange what they are saying is. This is often used for humorous purposes (even by the very character, if knowingly) or to add emphasis to how screwed up a life the particular character had, that they think something like that would be normal or conversation-appropriate.

Here's an example:

Old Lady: "Oh, this recipe? Well, when I fought the militias in Qurac, I met this woman who taught it to me. Very sweet lady, even though her husband made that Deal with the Devil. Anyways, first you take two eggs, separate the whites and the yolks... Are you listening?"

Similar to Bread, Eggs, Milk, Squick, only it's in a conversation and it's actually Bread, Eggs, Milk, Beat... wait, what? It can also be Bread Eggs Milk Squick depending on the content of the aside.

Film

 * Mortimer Brewster's aunts in Arsenic and Old Lace tended to speak much like this.

Live Action TV

 * Phoebe Buffay, from Friends, often begins a story with "When I was living on the streets..." or "When my mother killed herself..." - The titular friends take it in stride (after all, they already know about Phoebe's mother and life story), new acquaintances do not.
 * Also, when Phoebe was a Surrogate Mother she liked to freak people out by talking about having her brother's baby.
 * Chandler also had a penchant for this, especially regarding his father - though he was quite aware of how weird it was.
 * "The word you're looking for is, 'anyway...'"
 * Sue Sylvester in Glee frequently does this.
 * The Janitor in Scrubs does this sometimes.
 * The news segments on Top Gear often go off-topic for one reason or another, but no one derails them faster than James May.

Literature

 * Some of the wizards in the Discworld series have a tendency to make awkward pauses in the conversation even more awkward by filling them in with random trivia. In Reaper Man, the Senior Wrangler is said to be able to "do to a conversation what it usually takes thick treacle to do to the gears of a precision stopwatch."

Web Animation
"Sister: I thought about having a kid once. Tucker: Oh really? It's a lot of work. Sister: Yeah, it seems like it would be hard. But I thought, you know, who wants to be known as the girl who's had seven abortions? Tucker: Heh, yeah... wait, what?"
 * The character Sister from Red vs. Blue is known for this. She inserts all kinds of weird stuff into conversations that make people go "Wait, what?" as if it were normal.

Web Comics

 * Girl Genius had a curious moment when Her Undying Majesty Albia of England drops a really strange (and disturbing in most possible meanings) comment about a prospective addition to her garden right after Agatha left her unofficial audience. There are at least 3 explanations to it (could also be weird phrasing of something innocuous, but that's unlike her):
 * She may really be this weird. An ancient and godlike mind certainly could run projects about as weird even to Sparks as their own mad science ideas to common people.
 * Now that we know how the Second Breakthrough is achieved (at least in one variant, but she is secretive about any details of her own), it's clear that even after Mad Scientist insanity is not a problem, some part of her may be more alien than anything which fully belongs in good old 3D continuum. Being the only one, monarch or not, who uses plural for herself (even in a whispered aside to her daughter) fits with this possibility.
 * This may be a deliberate loyalty test for those still in her presence. She must know that Gil sort-of-turned her agent, and that one reason Heterodynes' minions don't think high of mind control, obviously, is Heterodynes being charismatic enough to induce Undying Loyalty without it (personal acquaintance with Agatha and composition of her retinue obviously illustrate this) — so this definitely could be a question in need of an updated answer for her.

Western Animation
"Spike: You know, Pinkie, these two ponies have a bit of a grudge match they're trying to settle, trying to prove who's the most athletic. Pinkie Pie: Yes! And grudge rhymes with fudge! Spike: Yes it... does... what? Pinkie Pie: And I like fudge! But if I eat too much fudge, I get a pudge, and then I can't budge. Spike: So...no fudge? Pinkie Pie: No thanks! I had a big breakfast. [...] Spike: Looks like Rainbow's doing her best to catch up! Pinkie Pie: I'm not sure how ketchup is going to help her in this contest. Now in a hot dog eating contest, it can make them doggies nice and slippery, but personally I prefer mustard. How about you, Spike? Spike: Uh...I like pickles?"
 * Pinkie Pie from My Little Pony Friendship Is Magic is seen doing this while doing the commentary for the Running of the Leaves in "Fall Weather Friends":

Real Life

 * Alfred Hitchcock did this in real life, especially as he was getting out of an elevator. Just as he was getting off at a floor, he'd turn to a friend and say, "So there I was with the dead body. Looking back I shouldn't have touched the knife, but, with such an amount of blood before me, I acted with poor judgement." As the elevator door shut, the other passengers would wonder what the hell had happened.
 * Ross Noble's entire routine is this. He begins practically every show by observing something strange an audience member is wearing or doing, and goes on to make a two-hour show out of it. A single show could cover Hare Krishnas, sailors, emus, and DJ Aslan before the night is out.