I dont think they died but it is possible to interpret that both died. Like for example one way could be that right after falling down, Jooha could have died and Dooshik imagined everything afterwards until he himself fell down on the ice and drowned, dying he saw his real Jooha. Him imagining Jooha losing memories is maybe a way for him to see Jooha happy and without any traumatic memories, also going back to the restaurant where they first met each other to feel closer to him. Haha depressing either way you see this ending…