"We told the court that we want to understand the events as to how the baby died and will interrogate Dharmishta Joshi to find it out," a senior police official said.
The police also told the court that they want to check if any other person is involved in the case.
Dharmishta's advocate Jagdish Hande argued that her police custody was not needed as the postmortem report was already with the police and they could investigate the case without her custodial interrogation.
According to police, Dharmishta allegedly beat up her baby severely causing her death.
While booking the mother on charge of murder, police claimed that the baby's father Kalpesh Joshi was innocent.
Ahuti died on September 30 during treatment after she was admitted by her parents in central Mumbai's civil-run KEM hospital with a skull fracture last week, police said.
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The hospital's paediatric department found that she had multiple skull fractures and also suffered from internal bleeding, police said.
When her parents failed to give a satisfactory explanation for her injuries, doctors notified the police.
However, after the girl died of injuries on Sunday, the police had registered it as an accidental death.
Later, preliminary postmortem findings confirmed that the baby had injuries on both sides of the skull, which were unlikely to be caused by an accidental fall.