The phrase "a completely different problem" is correct and usable in written English.
You can use it when discussing a situation or issue that is not related to the current topic or context, indicating that it requires separate consideration.
Example: "While we have made progress on the initial issue, addressing the budget constraints is a completely different problem that we need to tackle next."
Alternatives: "an entirely separate issue" or "a wholly distinct challenge".
I don't think one play illuminates the other: Hamlet is a completely different problem.
"That was a completely different problem," said Viñoly, insisting he was following a masterplan that specified arc-shaped towers.
Sondhi and Khemani were thinking about a completely different problem in 2015 when they worked out the theory of how time crystals could exist.
But this is a completely different problem from that of "legalizing" the presence of the millions who throng the favelas of Latin American cities.
But just eight years later, the city suffered its first blackout anyway when a completely different problem -- a fire in the Pearl Street station -- shut down the system.
It would be a completely different problem, though, if the bacteria turned out to carry a certain colistin-resistant gene called mcr-1.
It's to get the really talented people who are out there trying to build something cool and say, you know, if you joined Facebook, you could work on this completely different problem.
Finally, we applied MPINet to a drug-sensitivity dataset to test its validity with a completely different biological problem.
Instead of tinkering and revising to make incremental improvements in their established, successful innovations, a few great conceptual artists and entrepreneurs have switched to completely different problems, in spite of the possibility that they may fail completely in the new challenge.
Meanwhile Richard Lander had a completely different mobile phone problem, which happily has been resolved thank to quick action by EE. "In May, ahead of a holiday to Canada, I put money into my pay-as-you-go Orange mobile, giving me £126 credit," he told me.
But at first I didn't even realize it was the same question as a completely different-seeming problem that I had worked on for weeks!
