The name of the novel is "Whispers", written in 1990 by Dean Koontz.
Wiki site here:
serial killer, turns out to have been twins actually raised by their mother to believe that they were both the same person, in two bodies.
MATCH: Katherine gave birth to identical twin boys. The twins were born with cauls on their faces, leading the mentally unstable Katherine to believe they were demons. She raised her sons as if they were one person. They were both called Bruno, and both were rewarded or punished for anything either one of them did.
Somewhere late in the book one of the twins is killed,
PARTIAL MATCH (This actually occurs relatively early in the book): The next day Frye returns and attacks Hilary again, this time receiving several stab wounds before escaping. She calls the police and once again meets with Clemenza, who tells her that Frye's body has been found and takes her to the morgue to identify it.