Devinder Sivia was held on Wednesday night at his home in Oxford, after police discovered the body of Professor Steven Rawlings there.
After Dr Sivia was released yesterday, police said the death might be “a matter for a coroner’s inquest rather than a criminal court”.
A post-mortem examination proved inconclusive and further tests will be carried out over the coming weeks.
Detective Superintendent Rob Mason said: “This is a tragic incident and our investigations are ongoing to establish the cause of death. A substantial amount of information is already in the public domain and we can confirm that the two individuals involved have been friends for over 30 years.
“I would emphasise that the police are investigating all potential circumstances that could have led to his death. We are mindful that ultimately the death may be a matter for a coroner’s inquest rather than a criminal court and I would ask for patience from both the media and the public while we continue our investigation.
“Due to the post-mortem examination results proving inconclusive and further examinations being required, this has necessitated a lengthy bail date.”
Dr Sivia was bailed until April.