In the tradition I know this concept is not acceptable. Probably the key argument against purgatory has to do with questions of canon, that being that such a concept is not found in what is generally accepted as "Scripture." (The canon of Scripture is different within the different Christian traditions.)
Purgatory had a role to play in the Reformation. The sale of indulgences was what broke the camel's back for Luther and his theses spelled that out in Latin for the Church to ponder. The buying and selling of the forgiveness of sins was unacceptable, the practice being held up in relation to the understanding of Scripture.
Apart from these issues, what of purgatory itself? This is a question of the afterlife so it is not one easily defined. Nor is the thing in itself something that merits raising protest. Although questioned, especially these days, Heaven and Hell are commonly accepted among Christians. These are also afterlife questions. There is attestation throughout Scripture, no matter which canon.
From an earthly perspective, our experience bears out the popular notions of "hell on earth" or "heaven on earth," the former used more often, even among the irreligious. "Hell on earth" may be used to describe anything from war to devastating disease, unspeakable crime to widespread hunger. "Heaven on earth" may be a phrase used by the irreligious to describe anything that is considered good in the eyes of the beholder. "Heaven on earth" is most often understood within the wider Christian Tradition to describe those unseen realities that are related to the liturgy of the Church (be it Eastern or Western).
Heaven and Hell are different ends of the spectrum, whether speaking spiritually, experientially or theologically. Purgatory is somewhere in between, but in the direction of Heaven. Webster online defines purgatory as "an intermediate state after death for expiatory purification." If purgatory is a state after death, I would understand it as a divine work as it has to do with purification and is in the direction of heaven. Again, whether or not there is purgatory after death is not the focus of this post. I do ponder here the idea of a "purgatory on earth," apart from either the concepts of heaven and hell on earth and yet connected to both.
Rather than being caught in a constant high ("heaven") or a constant low ("hell"), most of us are somewhere in between ("purgatory"). This is like the Apostle learning to be content in whatever situation. Heaven and hell are happening on earth at the same time and they are happening to us. The Christian understands this in terms of sin and forgiveness and the gradual purging of the old life by the sanctification that comes with growing in grace. Both Scripture and experience bear out the realities of being purged as being refined by fire. The Scripture states that the faith is "more precious than gold."
We might say that just as there is hell on earth and heaven on earth that there is also "purgatory on earth". This could be understand as "expiatory"; the work of God such as we know in the passion and death of Jesus on the cross and being baptized both into His death and into His resurrection. We certainly have taste of such a state here on earth. He became flesh and dwelt among us that we might partake of His divine nature. As we follow Jesus and bear our crosses we come to know both the suffering and the joy of knowing Christ, who cleanses us from all sin.