I'm watching the first series of Luther and I can't understand why he has dropped the investigation into Alice Morgan. He knows she's the murderer, every other suspect he's relentless with (even bending/breaking the rules to catch them).
Why does he form an (almost) friendship with Alice instead of continuing the investigation - and why does the police force apparently abandon the case?
While the ultimate story arc (revealed in the last episode: see the previous question) retrospectively explains some of Luther's behaviour as far as the story arc goes, it doesn't explain Luther's motivation at the time. The nature of the somewhat complicated relationship between Luther and Alice develops over the three series and isn't really sufficient to explain what happens in episode 1.
There is a more obvious and more direct explanation of Luther's behaviour which happens to be consistent with his character and style of police work. In short, he can't pursue Alice Morgan because she has him over a barrel. The critical evidence Luther would need to ensure a conviction is the gun Alice used to shoot her parents. She hides it in the body of the dead dog which is cremated. Luther retrieves the cremated bones containing the metal parts of the dog but does so in an illegal way which Alice can prove. So Luther can prove she did it but can't pursue a conviction (any attempt to use the evidence would put him in jail and would not be usable against alice). So, when Luther confronts Alice late in episode 1 he destroys the evidence by throwing the created remains into the Thames.
But Luther has another motivation for this. He knows he can't take her to court, but he can hurt her pride in her perfect crime by removing her memento of the deed. He wants to undermine her arrogant pride in getting away with the crime and being admired for it. In the end Luther is happy knowing she is guilty and has destroyed her monument to her own cleverness.
Of course it all gets even more complicated after episode 1.
Updated: I'm going to put the answer in spoiler tags, but if you're watching the first series, really don't look
The last episode of the third series just premièred this week, and it's only just resolved your question, so I'd say just watch it and see, don't spoil it.
It's a 3 series story arc and fundamental to the whole plot, so inadvertently revealing it will spoil the rest for you.
So I'll say as little as I can
There is a mutual attraction between Alice and John which comes from Alice being one of the few people to understand his actions at various parts of the stories where he goes over the line. Initially she has escaped justice as although John knows she is guilty of her initial crime, the police are unable to prove it. She is arrested by the end of series one, but for what is part of the finale.
