Is it permissible to prostrate solely for supplication, by analogy with the action of Abdullah ibn al-Mubarak after completing the Quran?

It is not prescribed to draw near to Allah by a single prostration without a cause. However, there is no objection to a solitary prostration for supplication. Scholars have differed regarding absolute prostration without a reason. Some permitted it because it is an act of submission to Allah, and it is like the remembrances (dhikr) that are prescribed outside of prayer. It was narrated from Ibn Al-Mubarak that he liked, upon completing the Quran, for his supplication to be in prostration. This is not explicit that it was outside of prayer, as it could have been in the prostration of prayer.