This is nonsensical. Monads define a strict set of behaviors formalized as "monad laws"[0].
Perhaps what you want is a container which adheres to monad laws capable of abstracting exceptions. Two exemplars of same are Haskell's Data.Either[1] and Scala's Either[2].
0 - https://wiki.haskell.org/Monad_laws
1 - https://hackage-content.haskell.org/package/base-4.22.0.0/do...
2 - https://www.scala-lang.org/api/3.8.3/scala/util/Either.html
That is what I meant. Struggling to picture what the other "nonsensical" thing is.