Flowery Elizabethan English

The immense popularity of William Shakespeare and the King James version of The Bible has made the style in which those works were written very popular. For this reason, Flowery Elizabethan English is often the first thing that writers turn to when they want to show that a character is extremely old-fashioned—generally more so than an ordinary human could be. Their speech will be sprinkled with terms like "prithee" or "forsooth", and use obsolete pronouns like "thee" or "thou".

This is often used for immortals or near-immortals, like elves or gods, or for characters with a very strong connection to the era (perhaps a hyper-obsessive scholar). It can be used in alternate worlds and Fantasy works where there never was an Elizabethan England. May also be used by time travelers. Works written during or set in the Elizabethan era do not qualify, however, as the purpose there is quite different.

This even occurs in translated works, where it may signal a similar level of old-fashionedness in the original, or, in a language like Japanese, a formal or traditional style of speech that has no direct analogue in English.

In extreme cases, the characters may use Gratuitous Iambic Pentameter as well. When done badly, perhaps for humor, may shade into Ye Olde Butcherede Englishe. For characters who speak like they came from the much-later Victorian era, see Antiquated Linguistics. Talk Like a Pirate is similar, but quite distinct.

Examples of Flowery Elizabethan English include:

Anime and Manga

In Ranma ½, Tatewaki Kuno speaks this way in the English dub. This is a Cultural Translation, though, representing the archaic and poetic form of Japanese he speaks in due to his samurai delusions.
In Sekirei, Tsukiumi talks like this, most likely as a way of translating her formal Japanese. When she says "Have at thee, villain!", though, it's hard not to imagine her being Thor's Distaff Counterpart.

Comic Books

Thor, and all of the other Asgardians of the Marvel Universe, spoke until recently in Ren Faire-esque English. There have been several nods to Shakespeare over the years, including many quotes, mis-quotes, and even the character Volstagg the Voluminous, a parody of Shakespeare's Falstaff (from Henry IV parts 1 and 2). (The most recent relaunch of the character has him and his fellow Asgardians speaking formally but not archaically, and they keep their own font.)
Parodied in a comedy version of Alpha Flight, in which the Native American character, Yukon Jack, a loincloth-clad savage from the Canadian north woods whose tribe has had very limited contact with the outside world, speaks fluent Shakespearian all the time.
Much like Thor, Hercules and the Olympians from Marvel generally talked like this, too. This is averted and subverted at different times in the current run by Greg Pak and Fred van Lente. Hercules talks in modern English. When he goes to the Underworld at one point, his former human half talks in Shakespearean English. Hercules gets mad and asks why he talks like that when they're from ancient Greece.
In Empowered, the Caged Demonwolf combines this with Sesquipedalian Loquaciousness and Purple Prose (also, thesaurus abuse) for some truly remarkable dialogue.

Literature

In the Belgariad, Arendish folks talk like this, particularly the Mimbrates...though the Asturians deliberately change accents out of their contempt for the Mimbrates. One (non-Arendish) character trying to sound intelligent speaks like this for a few pages, before being explicitly told that she sounds ridiculous. Thoroughly and hilariously Lampshaded in The Malloreon when Poledra remarks that if they stick around the Arends long enough, everyone will be doing it. For his part, Eddings not only does the style grammatically, but (in The Rivan Codex) is highly critical of those who try but get it wrong.
Appears several times in The Elenium. All the speaking dead, whether they died centuries before or a few days before. A man playing a ressurected dead hero speaks this way, plagiarizing an old play. Also Bhelliom speaks this way.
In the Retief short story, "Ballots and Bandits", the natives of the planet Oberon all speak this way, for no apparent reason beyond Rule of Funny. (The name of the planet is a reference to the character from Shakespeare's A Midsummer Night's Dream.)
In Steven Brust's To Reign in Hell, most of the angels speak modern English, but Beelzebub speaks in a flowery Elizabethan flavor due to being injured by chaos:

"Rumors do fly about the land, milord. These have little truth in them. Whoso they light on taketh the worst o' the lie and sends that forth; whoso that lights on them doth likewise. 'Tis a most potent distillation of falsehood; milord, it will fall like the dew and make every angel drunk unawares."

In Roger Zelazny's novel, Creatures of Light and Darkness, a fantasy set far in the future, the immortal Prince Who Was A Thousand tends towards this style of speech, especially when conversing with his bodiless love, Nephytha. Other immortals and gods speak normal modern English, for the most part.
In The Dresden Files a number of immortals, particularly The Sidhe, have a tendency to use "thee" and "thou" in casual speech.
JRR Tolkien was fond of writing in an archaic style like that of the King James Bible.

Live Action TV

On Star Trek: The Original Series, the aged Vulcan matriarch T'Pau talks this way—presumably to show that, even by Vulcan standards, she's very old.

Newspaper Comics

In For Better or For Worse, someone who steals the door of Michael's dorm room does this when Michael asks where his door is.
In Calvin and Hobbes, Calvin begins imagining people talking like this in real life after being forced to watch a historical drama on TV.
Lampshaded in FoxTrot, when Peter decides to base his paper on Hamlet not on any of the countless thematic or symbolic topics it presents, but on the biggest question it raises of all: "What's with all the 'prithees'?"

Video Games

In Legacy of Kain, most of the dialog is Shakespearian speech, laden with archaisms and florid language.
- It should be mentioned that this series is an example of this trope working absolutely beautifully. The faux-Shakespearian script, highlighted by some of the best voice acting in video game history, is considered by many to be one of the best aspects of the series.
Grahf from Xenogears has a tendency to do this, along with a more general tendency to be a ridiculously Large Ham whenever he makes an appearance. "Dost thou desire the power?"
This was one of many, many jarring changes made to the King's Quest series by King's Quest Mask of Eternity. For seven games everyone's talk was very plain and modern, and then out of nowhere it's pseudo-Shakespeare city, even though this is supposed to be happening a decade or two later.
Frog in the original English release of Chrono Trigger on the SNES speaks with an Elizabethan dialect. In subsequent releases of the game, he speaks normally.
Cyan from Final Fantasy VI, while his speech isn't quite as fancy as Frog's (see above), also speaks in an old-fashioned manner, earning him the nickname "Mr. Thou" from Gau (which Gau sometimes mistakenly calls Sabin due to having met him at the same time as Cyan thus causing him to confuse the two).
The Great Deku Tree from The Legend of Zelda: Ocarina of Time uses flowery words and phrases such as "Thou hast verily demonstrated thy courage."
Intriguing example in Shadow of the Colossus. Dormin speaks a fictional language, but his lines are translated into English as verses peppered with 'thees' and 'thous'.

Webcomics

In Shortpacked, the Marvel Comics version of The Mighty Thor is parodied at the end of this strip.
The demon Skeezicks in Dandy and Company talks in bad pseudo-Elizabethan English. The cartoonist specifically made reference to Thor (see above) in describing his speech patterns.
Hibachi and the other dragons in The Inexplicable Adventures of Bob.
Minnie Mandy from Grim Tales from Down Below consistently speaks this way.

Web Original

On the Pokebattles parody site, both Green Valkyrie and John Mobius in Pokebattles Red Version talk like this (John was given a translator, for the audience and Lemony Narrator's benefit). That's probably why they fall in Love At First Sight.
In the Strong Bad Email "love poem", Strong Bad advises his fan to use this sort of language in his love poems, because "women love it when you get all Elizabethan."

Western Animation

The Simpsons, when they're at a Renaissance Faire.

Doris: Yon meat, 'tis sweet as summer's wafting breeze.
Homer: Can I have some?
Doris: Mine ears are only open to the pleas of those who speak ye olde English.
Homer: Sweet maiden of the spit, grant now my boon, that I might sup on suckling pig this noon.
Doris: Whatever.

Mr. Pricklepants from Toy Story 3. He is a thespian.
Dinobot from Beast Wars.
Princess Luna from My Little Pony Friendship Is Magic speaks this way in "Luna Eclipsed", having been imprisoned in the moon for the last thousand years. Surprisingly for a kids' show, it's all grammatically correct—not a stray "-eth" in sight.

Real Life

Brother Andrew (1928 - ) spoke like this when he was attending a missionary school in Great Britain some time after World War Two because he learned English by using a Dutch-English Dictionary and The King James Bible (first printed in 1611). In his autobiography God's Smuggler, he showed the effect this had on his English by recalling an incident where he once asked for butter saying "Thus sayeth the neighbor of Andrew, that thou wouldst be pleased to pass the butter." Oh, and he had a very thick Dutch accent that made it hard for him to pronounce the "th" digraph.