I can't find the exact qoute by Barth tonight, but I was told he said:
"Try to understand the Trinity and you will lose your mind; try to deny it, and you will lose you soul"
I thought that was a well-said version of many of my own thoughts on the matter, and applies well to many matters of God concerning predestination, purgatory, and literal Judgement calls.
Here I might add: I think of purgatory, not as a place, but as an instantaneous process of cleansing and firey division of goats from sheep, however that works. And it seems very possible every human being has some measure of goat, which must be torn out. On the other had, it is also possible there are people who are goats and people who are sheep, and there is only one or the other. I don't decide these things, nor do I know. (Oh, and the very idea of instantanous within the concept of eternity is silly.)
Wikipedia says that Barth rejected the label of neo-orthodoxy, and I can well understand why he might, although that is the way he is best known.
I also read on Wikipedia that he was an activist against the Nazis within Germany, which sounds like a very brave thing to have done.