Most murders are committed by someone the victim knows. That is often a spouse or relative, and they are definitely suspects.
Keep in mind, that what police want is evidence that can hold up in a court of law. And many murderers, if they're not caught 'in the act', get arrogant and think they've gotten away with it. Police know this, and play up to it, in order to get the suspect to talk. More people have talked themselves into jail than out of it. A suspect that is talking is one that will probably give away enough information to either link them to the crime, or absolve them. Either is good.
It's like weaving a net. Police carefully gather physical evidence, and also talk to everyone even remotely related to the victim, and to everyone who met/saw the victim shortly before their demise. If a suspect says, 'Oh, we bought ice cream at this booth'...then the police will be calling on that vendor.
Also, police prefer that suspects not 'lawyer up'. Attorneys usually advise clients not to volunteer any information, and insist on being present at any meeting with the police. Being accusatory towards a suspect is a good way for them to go running to a lawyer for help. If you talk to them politely, sympathize with the inconvenience, and dismiss any concerns about the suspect's involvement in the case; "Oh, you're not a suspect; but wouldn't you like to help find out who did this terrible thing?"...then the suspect is far more likely to cooperate.
Now, if the ex is suddenly playing Miss Marples or Nancy Drew...then yeah, the police will suddenly no longer be so friendly. For one thing, they can now arrest the person for 'interfering with an investigation' and 'interfering with official business'. It's a minor charge; but now the police have far more latitude for search warrants and questioning. And any friends accompanying the ex? They're suspects and possible accomplices.
So, yes, the ex/PC is a definite suspect. But it's very likely the police will pretend otherwise, at least until they have a reason to behave otherwise.