No, they didn't die. After that they got out of the water (confirmed by the author as well) . It's a nice reference to their "first" meeting (where they got close to each other) , when Doshik fell into the water and Jooha saved him, because Doshik can't swim. And here in the end, eventhough Doshik still can't swim, he jumped after Jooha, and as for Jooha the near death experience and the fall into the lake (the memory of their first meeting) made his memories come back.
I think it is a very nice ending, and not so rushed. They were shown to be happy: even with Jooha losing his memories, Doshik has done everything for him, and eventhough Doshik was a bit sad, he felt happy that Jooha does not remember all those bad things that happened to him.
And after he got back his memories, it implies that only more happiness is waiting for them
the ending is so unsatisfying, felt rushed, i wanted to see what really happened to them,i wanted to see them happy. Did they die when they fell in the frozen lake? i dont get it ╥﹏╥