Where did the idea of purgatory originate?

Not The Council of Trent.

Not Aquinas.

Not even Catholics.

But with the Jews, long before Jesus was born. Even today, Jews pray for their dead, in the 11 months of mourning, called the Kaddish prayers. Even if one refuses to accept the Book of Maccabbees as inspired, one cannot deny its historical value. They prayed for the dead.

Purgatory was a belief of the earliest Christians including Church Fathers such as Tertullian, Clement of Alexandria, Origen, Cyprian, Jerome, Ambrose, Augustine (between 200-500 A.D.) Prayers for the dead were inscribed on the tombs of the early Christians buried in Catacombs. Why would they need prayers if they were in Heaven? There is no evidence in early Christian writings that there was a struggle with this belief, it was accepted and well known. Purgatory was not invented at the Council of Trent. It was indoctrinated then because it was being called into question for the first time.

Purgatory is for the saved.

Purgatory does not pay for sin, Jesus on the cross payed for our sins. Purgatory pays for the consequences of our sins.

If I throw a rock through your window, and repent of that sin, and I am forgiven of that sin by Jesus' death on the cross, Jesus does not pay for the window. That is my responsibility or my repentance is meaningless. Purgatory pays for all the broken windows of our life that we never got around to paying. It is not about paying for sin.

Some Christians reject this basic doctrine that was accepted by the entire Christian world for 1500 years, until the introduction of "justification by faith alone", a theological opinion of the middle ages.

I believe it was Thomas Aquinas. He logically deduced that if you die with unconfessed sins, there needs to be a way for you to make atonement for these sins. You are then purified through purgatory. You could get more info if you look up Aquinas.

the idea is interesting

and we share the same belief in Islam in the following sense:

when people who have even the smallest bit of faith in God die as sinners, they are first sent into hell after theyre judged on judgement day, then when they have paid the consequences of their sins in full in hell, they are finally allowed to enter heaven clean without sin, this is because they have faith nevertheless.

and this shows the extent to which God is merciful with His creations.

And God knows best

Catholics believe this. There is no purgatory - just Heaven or Hell only - nothing else.

Like God, the Bible etc. It's a man-made myth used to scare the gullible into the belief of a supreme creator.

Man made by Thomas Aquinas.

The Bible is NOT man made it is the insired (God breathed) word of God.