Murderers should be able to go after the detective.
this would be one of the few murderers, a person who goes to kill someone, notices the detective is on to them and then is set to kill the detective
this would happen after the second murder they would leave a note stating they know that their being tracked down and the player would be advised to "stay home" or they'd be told that "your next"
the murderer wouldn't know exactly who we are but they'd search for us through their daily life while trying to stay low, if they see you in public they might follow you around for a little before dropping it, sleeping at night? They might break in and shoot you in bed, in a bathroom washing up, they might stab you in the back. Shopping at a store, it just so happens that they want to shop at the same store. Going home they might just follow you and wait outside your door until your distracted.
Leave your door unlocked? Guess whos waiting...
Seeing something along the lines of this would be pretty nice and would add more of a paranoia factor, I like to personally roleplay my characters as paranoid 24/7 due to their line of work and the people they catch