Barth’s concern is precisely to understand the ancient term “hypostases”. He points out that, since the Enlightenment, the word “person” has come to mean something like “individual centre of consciousness” (which it never meant in patristic theology). So he suggests that we should stop talking about the trinitarian “persons”, and should instead use the term “modes or ways of being” (which Barth very carefully defines on the basis of the ancient term “hypostases” — and Barth’s whole trinitarian theology opposes modalism).

Not all theologians have agreed with Barth that the term “person” should be dropped; but it’s widely agreed that “person” now means something very different from what it meant to the ancient church: the creeds certainly aren’t saying that God is three individual centres of consciousness!

Ben Myers on Karl Barth’s reconsideration of using the term ‘person’ to describe the trinitarian ‘modes of being’ with the disclaimer that it was used in the comments section, so probably wasn’t meant to be quoted, but I thought it was a good summary of this interesting argument in Karl Barth’s Church Dogmatics I/1