I don't believe the girl is a simple hallucination - not unless this is an extended prologue for a different story, or the writer is a hack. There's too many signs that point to her existence.
The police officer has a picture of her; the noose wasn't where it was meant to be; the fat guy clearly walked up and responded to something while he was gone; and something ruined his book in this chapter. This is all clear evidence for her having been identifiable by people in the past, and of something other than the guy having an influence on the world around him.
Now, some of these things could have a mundane explanation. Perhaps the noose doesn't exist either, perhaps the fat guy was just thinking of stealing his phone, and perhaps he ruined the book himself and had another mini-breakdown while his broken mind concocted another explanation for what happened. But those go beyond a simple Sixth-Sense "actually something else was going on the whole time!" twist, and straight into "The author was actively lying to you while wasting multiple chapters doing nothing but distract you from the obvious conclusion" territory. Simply put, there would be no reason to explain these points if the author hadn't chosen to include them, and there are much more subtle situations one could use if he was simply meant to be crazy.
Especially since we actually saw things from her perspective for a brief period. That would be a massive violation of author-reader trust if she straight-up didn't exist, since that couldn't be explained as him hallucinating. That would be the author just directly writing an exchange that never happened.
That said, it does seem clear that she's not corporeally there. Her existence in the car at all doesn't make any sense if that were so - it breaks credulity that, as he ran off pursued by co-workers, that she could or would follow and hop into the car without him noticing. Similarly, playing hide-and-seek with the cops, or the way fatso never tried to justify himself or ask for her help in explaining things... She could just have a bad case of Manic Pixie Dream Girl going on while nobody else bothers to talk to a kid, but it's pretty unnatural if that's so. Plus; the diary.
Which leads me to my own personal conclusion; she's some kind of malevolent supernatural entity. The guy has a sudden, severe deterioration in his mental condition, as he ends up involved with a girl that nobody else interacts with... Combined with the drawing in his journal of a girl with a horrifying face, drawn on a field of black. That sounds consistent with her eating his SAN points to me. I mean, what kind of person just hauls off and assaults their boss before running off to kill themselves instead of just... Quitting? Just not showing up for work one day? I mean, he wouldn't have to worry about the cops interrupting his pitiful little bucket list ("go to the beach and reread your favorite book"? Really? The guy certainly aims low... "Learn to play the guitar" was the only thing on that list that wouldn't fit into a lazy weekend) if he just hadn't punched his boss. And nothing else on his list seems consistent with that kind of violent assault.
It's possible that this relationship is reversed, granted; perhaps he witnessed something terrible happen to her (or did something horrible), and that's her lying in anguish on a pool of blood... But he never saw her outside his breaks. When would he have heard about her murder? And would that really have led to this kind of slow breakdown instead of acting as the breaking point on his already stressed life? Unless, of course, he did stalk and murder her himself, and this is him breaking down from lack of relief... But, well, it doesn't seem consistent with almost any of the rest of his personality. His unnecessarily violent outburst, perhaps, the fact that he can't remember how he started talking to the girl could suggest he's suppressing other things, and the clear difference between him and how the witnesses perceive him... But it seems too early to suggest that he could be a murderer.
But if she isn't at least (in either direction) a ghost on the sliding scale of "real to not real", then I'll admit to being a bit disappointed and confused by some of the writing decisions.