The Monk's TARDIS was a Mark IV Type 40 TARDIS (PROSE: A Brief History of Time Lords). It was a more advanced model than the one used by the Doctor. According to the Doctor, the Monk had stolen his TARDIS fifty years after he'd left Gallifrey.
Unlike the First Doctor's TARDIS, when the Monk and the Doctor first crossed paths in their respective careers as reneagdes, the Monk's TARDIS had a functional chameleon circuit, directional unit and an automatic drift control. Its control console was also set on a dais, as opposed to being at floor level like other TARDISes. (TV: The Time Meddler)
After visiting several moments in Earth history, where he collected valuable pieces of art, meeting Leonardo da Vinci where the Monk discussed the possibilities of powered flight with him and made a fortune in compound interest by leaving money in a London bank in 1968 and travelling forward in time 200 years, he landed in Northumbria in 1066 and disguised his ship as a Saxon sarcophagus in an abandoned monastery. The Doctor learned of the Monk's plans to change history and sabotaged his TARDIS by taking the dimensional control, making it the same size inside as outside. This made it impossible for the Monk to get inside it.
The Monk eventually managed to fix the dimensional controls, re-creating the interior dimensions, and thus escape from 1066. When the Fifth Doctor next encountered the Monk, he appeared to have voluntarily re-configured his TARDIS into a police box. The Monk had the top of his ship's roof blown off by Ice Warriors with a sonic cannon. (COMIC: 4-Dimensional Vistas)
On a later adventure on the volcano planet Tigus, Mortimus's ship turned into a large rock. Meeting again in ancient Egypt, the First Doctor changed it into a police box so Mavic Chen and the Daleks would mistake it for the Doctor's TARDIS. During the Doctor's reprogramming of the chameleon circuit of the Monk's TARDIS, the ship briefly took the forms of a motorbike, a Western covered wagon and a World War II tank. He also stole the Monk's directional unit so the Monk would re-materialise in a random location. The Monk ended up in a frigid planet with his TARDIS disguised as a block of ice. (TV: The Daleks' Master Plan)
The Seventh Doctor met the Monk in London in 1976. Mortimus, posing as a record executive, had turned his ship into a desk, complete with scraps of paper. The papers retained some of the special qualities of his TARDIS, allowing the Doctor to use them to return to Earth after the Monk vengefully tried to abandon him on the same icy planet where he had been trapped. At this time, the Monk had reconfigured his TARDIS's interior into a gothic style with icicles to remind him of the ice planet he'd been stranded on. (PROSE: No Future)
The Eighth Doctor, along with his new companion Tamsin Drew, had a reunion with the Monk in the Abbey of Kells in Ireland in the year 1006. Under the alias of Abbot Thelonius, the Monk used the impressive artistic skills of the monks to draw a new directional unit for his TARDIS, using a page of the Book of Kells and liquid Gallifrite. Unknown to the Monk, Tamsin had switched his circuit for a false one, prior to the first test of the circuit and again the Monk's TARDIS was crippled. However, the Monk was able to rectify this and had several more meetings with the Doctor. (AUDIO: The Book of Kells)
When the Monk took Tamsin to the planet Halcyon in the 33rd century, his TARDIS assumed the form of a Punch and Judy tent. The Monk told her the TARDIS was his pride and joy, and despite the damage it had sustained he claimed it only needed some polish before looking as good as new. The Monk also observed as the TARDIS materialised on Halcyon that the Mark IV had a smooth materialisation action, and boasted he knew where he was and when compared to the Doctor. The Monk told Tamsin that it could have taken the form of a tree which would have made it less conspicuous, but that would also have made it more difficult to find, given that it had materialised in a forest. (AUDIO: The Resurrection of Mars)
Working with the human/Hetroden hybrid Sophia, the Monk was able to trick the Fifth Doctor into allowing history to be changed, giving the Monk the chance to step into the 'hole' created in the timeline when the Time Lords took the Doctor away as punishment for his violation of the Laws of Time. As a result, the Monk began to travel in the Doctor's TARDIS, assuming the Doctor's role in history. However, realising that the Monk's actions were wrong, Sophia was able to draw the Fifth Doctor back into existence, allowing the Doctor to use the Monk's original TARDIS to set a trap that forced the Monk to restore the Doctor to his rightful place in history. (AUDIO: The Secret History)
At some point, the Monk did away with the TARDIS laboratory, something which appalled the Third Doctor when the Monk's interference with human biology warranted it. The Monk's TARDIS did, however, have a fully equipped home cinema, and several kitchens. (AUDIO: The Rise of the New Humans)
The Monk used his TARDIS, disguised as a databank, to open a black hole linked to a pocket dimension. The Second Doctor and Pavo managed to counter the actions of the Monk, which resulted in him falling into the pocket dimension. (AUDIO: The Black Hole)
When the Last Great Time War broke out, the Monk, fearing for his life because he knew the Daleks would kill him, ran and travelled to Earth during Tudor times. After pushing the TARDIS, the Monk was overzealous in removing components so that neither the Time Lords and the Daleks would find him hiding away, badly damaging his ship in the process. The Monk's generous removal of parts eventually stranded him on Earth but he managed to rig an opening into the Matrix to keep an eye on the war.
Seeking to attract the Time Lords' attention, the Monk began meddling with history by adapting his chameleon circuit to appear as Henry VIII to attract the Time Lords attention.
Unfortunately, Missy found him first and managed to manipulate events so she gained access to his TARDIS to steal a component for her Vortex Manipulator, stranding the Monk on Earth. The Monk was even willing to abandon his TARDIS in favour of escaping by using the Vortex Manipulator, but Missy was able to escape first and get the better of the Monk, leaving him stranded and swearing revenge on her. (AUDIO: Divorced, Beheaded, Regenerated)
The Monk eventually managed to repair his TARDIS and went searching for Missy to exact revenge on her. After he'd caught up with her, the two were taken prisoner by the Ogrons and taken to their homeworld as prisoners, the Monk's TARDIS, still disguised, being placed in storage. When Missy sought to use the TARDIS to escape, the Monk revealed that, tired of others stealing pieces of his ship, he'd locked the controls to only respond to him. After the two had managed to flee the Ogrons, Missy found the ship and made the lock only respond to her, forcing the Monk to take her with him. (AUDIO: Too Many Masters)