Byzantine Catholics do not hold the doctrine of purgatory. We respect it as a valid approach but it is not ours. Byzantine Christians teach that there is a state of preparation (or purification) after death and that prayer for those in this state is helpful. The image used by the Church Fathers is one of purifying ascent to the Father.
Which, by the way, is what the Catholic Church as a whole teaches.
Dantesque imagery about purgatory is that, mere imagery and not to be mistaken for doctrine.
Since we are all sinners, and since nothing sinful can enter Heaven, it follows that between here (inclusive) and there (non-inclusive), all those who have received the gift of Salvation in Christ must be purified from all sin.
That is, that all sin must be purged from themselves.
Deuterocanonical Scripture (the books of the Macabees) are quite explicit that this could happen after physical death.
But also, the Gospel itself implies that, when Jesus says that the sin against the Holy Spirit cannot be forgiven in this age, nor in the age to come. The remark would be very puzzling if there were no sins that could be forgiven in the age to come.
And so, the Catholic Church teaches that there is "something" between earth and Heaven where the last remnants of sin can be purged, or purified. Or, in Eastern terms, where the final step in theosis can take place.
I like the imagery used before on this forum of being in a dark room and coming out to the bright sun light. Your eyes would need some time to adjust, and yes, it hurts, but the pain goes away soon and it is quite worth it.
The "darker" your life is, the harder it would be for you to get used to the light. If your life is full of light, then that final step could not even be necessary, although I personally believe that this is a very high mark, for even the holiest men and women, while on earth, are exposed to the powerful evil forces that roam this world.
As a Western Catholic, this is what I believe about purgatory. Is this so far away from the Eastern understanding?