Batman

Batman, also known as the Dark Knight, (TV: The Return of Doctor Mysterio) was, according to differing accounts, either a costumed comic book superhero on Earth. (COMIC: Moving In, etc.) or a real individual who travelled with Gandalf and Wyldstyle. (WC: New Adventures Await!)

A version of Batman existed in Earth-33⅓, (COMIC: Doctor Who? 64, Doctor Who? 138) where he was apparently a possible incarnation of the Doctor. (COMIC: Doctor Who? 138)

Biography
After travelling through the Vortex with Gandalf and Wyldstyle, the trio landed on the streets of New York City, where Slimer flew past. The Ghostbusters quickly began attempting to capture the ghost, before it broke free, with Harry Potter having to help. The trio then ran past Ghostbusters H.Q. and into a portal.

Reuniting with Gandalf and Wyldstyle, he then found himself in the Land of Ooo, before Finn and Jake appeared, proclaiming "It's battle time!". He then crashed into a cave where he met Sloth, asking "what took [him] so long" before exiting the cave.

He later charged into battle alongside others, before ending up in a mosh pit including everyone who participated in the battle. (WC: New Adventures Await!)

Other references
According to Maxwell Collins, Batman was "the world's greatest detective", and despite having no powers, he could beat "anyone". The Twelfth Doctor didn't believe that "solving crimes after they happen[ed]" made him clever, let alone cleverer than Captain America, and thought that Batman was a bored billionaire with "too much time on his hands". Batman wielded "a batarang" [sic], and fought what the Doctor referred to as "an idiot in clown make-up". Batman had a "cool car", protected Gotham City, and avenged his deceased parents. Maxwell thought that Batman would win in a fight between Batman and Captain America, while the Doctor favoured Captain America. (COMIC: Moving In)

In one issue of a comic, he was faced up against Superman, and, as the cover promised, "only one can survive!" (PROSE: Curiosity) Batman was featured in at least one issue of Detective Comics. (TV: The Return of Doctor Mysterio)

He had a butler named Alfred (AUDIO: A Kill to a View) and a sidekick, Robin. (PROSE: In the Sixties) Sarah Jane Smith knew that he "only hired orphans as sidekicks", mentally comparing it to the Doctor's preference for largely-unattached companions. (PROSE: Interference - Book One)

During a visit to Shoreditch in November 1963, Ace was wearing a Batman earring in her left ear. (TV: Remembrance of the Daleks)

In the 1970s at the Inferno Project, when Greg Sutton expressed surprise at the appearance of the Third Doctor's TARDIS console, the Doctor sarcastically asked Sutton whether he'd expected "some kind of space rocket with Batman at the controls" instead. (TV: Inferno)

Stuart Hyde playfully called Dr Ruth Ingram "Batman". (TV: The Time Monster)

Ace once called the Seventh Doctor "Batman." (AUDIO: Night Thoughts)

When the Twelfth Doctor and Clara Oswald landed at the San Diego Comic-Con in 2015, the Batmobile from one of the Batman films was being shown off. (COMIC: Selfie)

In the far future, Miranda Dawkins occasionally wore a Batman T-shirt. (COMIC: Miranda)

Clyde Langer jokingly referred to Sarah Jane's attic at 13 Bannerman Road as "the Batcave", after Batman's secret base of operations. (TV: Sky)

Upon completing his own superhero comic book, The Silver Bullet, Clyde declared, "Step aside, Batman!" (TV: The Curse of Clyde Langer)

St John Colchester once described himself as a better-dressed Batman. (AUDIO: A Postcard from Mr Colchester)

Behind the scenes

 * David Tennant has expressed interest in playing popular Batman villain the Riddler in a future Batman film.
 * , who played several roles in , was considered for both the Eighth Doctor and in the 1996 TV movie. He was also considered for the role of Sharaz Jek in the 1984 story The Caves of Androzani.
 * , who played the Riddler in The Batman, was considered for the role of the Master in the 1996 TV movie.
 * At a charity stage show in Hollywood, Matt Smith played a spoof Batman in Gotham Autopsy, a comedy sketch written the night before.
 * A persistent rumour states that Jim Carrey, who played the Riddler in Batman Forever, turned down the role of the Eighth Doctor as he had never heard of Doctor Who and thought that it would cause outrage amongst fans if the role were played by a non-fan.
 * Batman appears as one of the main characters in LEGO Dimensions.
 * The Daleks make an appearance as major adversaries in The Lego Batman Movie (though, as part of a joke, they are not referred to as Daleks but merely "British robots" as fourth-wall-breaking from villain the Joker).
 * Sean Pertwee, who plays Alfred Pennyworth in Gotham, is the son of Jon Pertwee.
 * Jo Martin played a police officer in.