An out-there theory that actually has supporting evidence: Arthur and his mother, Penny, are the same person!
As far as I know, I'm the first to notice this. Please give credit if you spread it.
Penny is Arthur's alter-ego. Penny transitioned to Arthur during her/his stay at Arkham Asylum. In Arthur/Penny's disturbed mind Penny transformed from being his former self, to becoming his mother.
Supporting evidence:
1. Arthur and his mother are never seen together by anyone else throughout the movie.
2. One of the bigger clues takes place when the two detectives approach Arthur outside the hospital where his mother is supposedly being treated after a "stroke".
The officers mention that they spoke to Arthur's mother because they couldn't find him. Possibly they spoke to her on the phone. They don't mention that they actually saw her. They say that upon speaking to her she began hyperventilating and fell and hit her head badly. She didn't actually have a stroke, this was a lie by Arthur.
Notice that the entire discussion with the officers is one lie after another. Afterwards as he tries to enter the hospital he mistakenly tries to enter the hospital through the exit. This might be a clue that he never went into the hospital before this. If he had come out of that hospital he would have remembered the exit.
I suspect that Penny's hyperventilating and hitting of the head that the detectives mention did actually take place. After Penny's (and Arthur's) head injury, the restraints on Arthur were removed because Penny's ego became silent. Later Arthur kills Penny and fully transforms into Arthur.
3. The confrontation between Arthur and Thomas Wayne at the Charlie Chaplin show is imagined. The beginning of the dream sequence mirrors the dream sequence where he meets the late night talk show host. He is looking at the protests on TV and suddenly he's there. The events leading upto the meeting with Wayne are very unbelievable. The dream sequence ends with him in the apartment standing with arms on kitchen counter mimicing his arms on the sink in the dream.
Both the "audience priming" dream sequence with the talk show host and the Wayne dream sequence end with heavy blue lighting in the apartment.
4. During the conversation with the clerk in Arkham, he clerk mentions "She was found guilty of endangering 'her own' child". The adoption paper was discovered by Arthur later, he is already primed to see this illusion due to his made up dream confrontation with Wayne. The adoption paper is likely not real.
Assuming that the clerk is reliable, Penny/Arthur did have a child. It's unbelievable that the child would be sent to Arkham. Possibly the child was taken away from her and put up for adoption and she became Arthur as a coping mechanism for losing her child.
It is doubtful that Penny actually worked for Wayne. Alfred probably knew about her due to news surrounding her incarceration. She built up an illusory strong attachment to Wayne just as Arthur built up an illusory strong attachment with both the TV show host and Wayne (again, as his new self). As Penny, Wayne was an illusory lover, as Arthur, Wayne became an illusory father. The psychological root of this is probably from a desire to have a wealthy happy life.
5. A couple other hints: Before killing Penny, Arthur mentions "I always hated that name." Referring to the name Penny Fleck. A possible reference to the idea that it was his own name and he prefers his new name, Arthur.
Arthur and his mother shared a bedroom. During the scene prior to the killing of work friend, he is putting on white makeup infront of what clearly looks like a female's dressing table (Penny's/his own). He takes a picture of young Penny and crumples it symbolizing that after killing Penny he has become entirely Arthur.
6. When Arthur is speaking to the social worker early in the film he is asked "have you thought more about why you were locked up?" to this Arthur responds "who knows". It is clear to me that Arthur was locked up as Penny. She transitioned during her stay there, possibly as part of her treatment.
Also, remember this: When he is talking to the Arkham clerk he asks something like "How does one end up here" as if he wasn't there already. Infact, he was there before as Penny. As he is losing his Penny-self due to lack of meds and also possibly due to head injury, he is forgetting his old self. Arthur was not in Arkham, Penny was.