Option 1: at cross purposes -> means people are misunderstanding each other because they're talking about different things or have different objectives.
Option 2: to some purpose -> means with some useful effect or result, doesn't fit the context of misunderstanding.
Option 3: in the dark -> means without knowledge or information, doesn't describe the miscommunication happening here.
Option 4: through our hat -> means talking nonsense or without proper knowledge
Hence, Option 1: at cross purposes → This idiom perfectly describes the situation where two people are having a conversation but are actually talking about different things, leading to confusion.
Option 4: through our hat → Also acceptable, as it implies speaking without proper clarity or understanding, which can result in such confusion.