The Monk's TARDIS was a Mark IV TARDIS. It was a more advanced model than the one used by the Doctor.
History
Unlike the First Doctor's TARDIS, at this time 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. After visiting several moments in Earth history, where he collected valuable pieces of art, 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. (TV: The Time Meddler)
The Monk eventually managed to fix the dimensional controls, re-creating the interior dimensions, and thus escape from 1066. 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 an frigid planet with his TARDIS disguised as a block of ice. (TV: The Daleks' Master Plan)
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)
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. (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 circuit 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)