Amid the stress and sleeplessness of watching their little baby in intensive care, things were starting to look good.

So much so that Stephen and Hannah Lavery were encouraged by hospital staff to take a night off and attend a friend's wedding. Their son Alexander, they were assured, was doing well.

Alexander had been born 15 weeks prematurely in February last year, when Ms Lavery went into a random and unexplained labour.

Despite being born so early, he was given an 80 per cent chance of survival by specialists at The Royal Women's Hospital. And with every week Alexander spent in the hospital's neonatology intensive care unit, his chances of survival increased.

But within days of his parents attending that wedding in country Victoria, Alexander's condition deteriorated and the progress he had made was lost when he went into cardiac arrest. The life that began so early lasted just 4½ weeks.

Over the difficult weeks that followed, Alexander's parents began to accept that the odds against their son – who was born at 892 grams – were just stacked too high.

So, five months after his death, to be told by hospital management that the neonatologist's clinical decisions contributed to Alexander's death came as a bombshell.

"We had just started accepting that our baby was born too young and was too premature to survive," Mr Lavery said.

"To then be told it was the hospital's fault and the back and forth we've had with them since, it's been quite angering."

The Royal Women's reviewed Alexander's case and found the neonatologist responsible for his care made poor clinical decisions over the removal and then insertion of the baby's breathing tube.

The doctor denies she was responsible for Alexander's death and launched legal action, arguing the hospital's moves to sack her breached an enterprise bargaining agreement.

The case went to the Federal Court and on Monday mediation between the doctor and the hospital finalised. It is unclear whether the doctor is still employed by the hospital. She was on paid leave.

The Laverys said the extubation and intubation procedures, along with their son being put on steroids, were done over four days.

They have since learned the process should have instead been more gradual and stretched over between 10 days and a fortnight.

The neonatologist was eager to get Alexander off the ventilator, they were told, out of concerns a reliance on the breathing machine might cause long-term lung disease.

After they were later told of the findings in the hospital's review, Alexander's parents became concerned by the number of clinical decisions the neonatologist made without being challenged by colleagues, and that the hospital lacked a clinical review process.

The couple's concerns – and their dealings with the the hospital – prompted them to request the coroner investigate, out of misgivings the hospital has failed to address systemic problems and there is "every chance it could happen again", Mr Lavery said.

The case is now before the Coroners Court.

Mr Lavery said he and his wife were shocked to learn this month that the hospital also found the neonatologist responsible for another baby's death, six months before she treated Alexander.

The doctor also rejects the hospital's finding on the other baby.

"You want to make sure they are held accountable for the decisions they made," Mr Lavery said.

"But I don't want her to be made a scapegoat and for them to say 'we have got rid of one rogue doctor, because she made decisions on their behalf as an employee of the hospital'.

"They need to do everything in their power to make sure this sort of thing doesn't happen again."

The Royal Women's Hospital declined to respond to a list of questions.

Chief executive Sue Matthews said she could not comment because the case was before the courts, but expressed her condolences to the family.

Before mediation, the hospital alleged the neonatologist poorly planned the extubation process and that her clinical decisions were inconsistent with her continued employment.

However the doctor's lawyers argued the hospital had previously told her it couldn't rule whether her care or clinical decisions contributed to the babies' deaths.

The doctor had argued her reputation and health would suffer irreparable damage if she was sacked, and that the hospital had not provided adequate support to help her manage her depression.

The hospital previously said it received complaints about the neonatologist from colleagues, including allegations of confrontational and demeaning conduct towards other medical staff.