These moments are rather uncharacteristically vague for Shakespeare – and for Hamlet. “So much for this” (What was the this?) and now for “The other”?
We can guess at what these two things are referring to but there isn’t any real evidence for what the this and the other are referring to.
Perhaps this is to suggest an intimacy between Hamlet and Horatio wherein they share private chat that even the audience is not privy to. They can speak in code. The way that the Holly Hunter and Albert Brooks characters do in Broadcast News.