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Max was twice offered a chance to travel with the Doctor in his TARDIS. He refused both times, preferring instead to run the Stockbridge Division of the Bureau for Interplanetary Liaison, a group of people extending a welcoming hand to visiting aliens. He was, in essence, the Stockbridge "UFO chaser", viewed by his neighbours as a crackpot.
Practically the only resident of Stockbridge who believed in his cause was Izzy Sinclair, who joined his organisation. It was through Max that Izzy was introduced to the Eighth Doctor and became one of his longest-serving companions. (COMIC: Endgame)
Max met the Twelfth Doctor again during Max's sixtieth birthday in Stockbridge in 2016. Max discovered that the villagers had been frozen in time; the Doctor explained that Max was unaffected because he had flown in the TARDIS before. After Max was captured by Gol Clutha, the Doctor and Max were taken to Josiah W. Dogbolter inside St Justinian's Church. The being Dogbolter believed had frozen Stockbridge to lure the Doctor into a trap was in fact the shapeshifting WhifferdillFrobisher, and not Chiyoko as he originally thought. After Frobisher had incapacitated Dogbolter's henchmen, Max punched Dogbolter, saving the Doctor's life. At the same time, Izzy arrived at the church, knocking Dogbolter out with the complete X-FilesBlu-ray. Dogbolter was subsequently arrested and his assets seized when Sharon Allen caught his confession to murder on camera. Izzy realised that the Doctor made a trap for Dogbolter, and made Dogbolter think that it was Dogbolter's trap for the Doctor. The Doctor also explained that he froze the villagers as "the simplest way to keep everyone safe". To celebrate Max's birthday, the Doctor took Max and some of the Doctor's friends to the city of Cornucopia. (COMIC: The Stockbridge Showdown)
Behind the scenes
Maxwell Edison's name comes from the Beatles song "Maxwell's Silver Hammer". This was referenced in-universe in the audio story The Eternal Summer, when Lizzie Corrigan pointed out the connection. He joked that he was lucky — he was originally going to be called Eleanor Rigby. This must have been a coincidence in-universe, as Max would have been thirteen years old by the time Abbey Road was released.
Although (originally) a comics-only character, Max had a certain degree of commonality with the televised character of Wilfred Mott.
Multi-adventures reference the rare instance when a companion is shown or stated to have multiple adventures with the Doctor but only appears in a single story.
If a medium is not mentioned, then this incarnation did not have companions who were original to that medium; it does not mean that this Doctor failed to appear in that medium.