Sorry for the additional pedantry, but I think this important to be precise about given the target audience of your comment.

Monads aren't the separation between purely functional and stateful code. The Haskell type system maintains that separation. Anything that's doesn't return IO a for some a appears to be a pure function from the perspective of the programmer. Once a function returns IO a, there aren't any* functions provided by the compiler that can make a function that uses those results not also return IO b for some b. For example, the type of getLine is IO String (because it impurely produces a String) and the type of putStr is String -> IO () (because it takes a String and mutates the world without returning anything).

If the compiler provided a function for computing on the a in the IO a, for instance, bindIO :: IO a -> (a -> IO b) -> IO b and a function to wrap the results of non-IO functions, such as returnIO :: a -> IO a, you could do arbitrary computation with these IO-wrapped data types, but know at a glance if your functions were impure.

This approach doesn't require the Monad typeclass at all, just a magic type called IO that tags impure computations that are implemented with compiler and runtime magic. It happens to be the case that this is exactly how GHC implements the IO type. bindIO is implemented here[0] and returnIO is implemented here[1] and the compiler magic used to implement them isn't* exported, so all IO operations have to go through those functions. It is not a coincidence to that these functions have the right types to form a Monad instance for IO and indeed, that is also present[2], but the IO type and the type system that ensures it can't be sneakily hidden are doing the heavy lifting, and the Monad instance (and accompanying syntactic sugar), are just there to make it nicer to work with and easier to abstract over.

If you have a passing familiarity with Haskell, the phrase "state monad" is the obvious place where my claims stop making sense. In fact, the State type only supports computations that are entirely pure. If you want to simulate global variables in a language that didn't have them, you could always pass all of your global variables to every function and get updated ones back from the function along with the nominal results of the computation. The State type is just a regular data type that wraps stateful functions constructed by such state passing. A type of the form State Int String is just a function that takes an Int and returns and String and an Int, no compiler or runtime magic needed.

You can play the same trick as in the IO case and provide functions bindState :: State s a -> (a -> State s b) -> State s b and returnState :: a -> State s a in order to compute on these "stateful" values while making sure the result state got passed to the next function in the chain correctly. Like IO these two functions can be used to create a Monad instance for State. Unlike IO, State is just a data type holding a regular Haskell function, so it's extremely reasonable to write a function of type State s a -> s -> a which runs the State s a computation with an initial value of type s. This is written by unwrapping the State type and then passing the initial state value to the function inside and return the result while ignoring the returned new state. More details on how State is implemented are available here[3].

A complication to this is that if you want stateful mutation for performance reasons, the ST type[4] also exists, which looks identical to the State type from the programmer's perspective, but plays similar tricks to IO in order to actually mutate under the hood while not exposing the implementation details to the user, so it can be reasoned about exactly as if it was pure and using the same implementation as State.

These Monad instances for IO, State, and ST start to pull their weight when you write functions that only use features provided by the Monad typeclass and they work seamlessly with any implementation of stateful computation despite their very different internals. Monad is quite general, so if all you care about is abstracting over stateful computations, you can also use the methods from MonadState[5] which allow you to interact with the state along with the results of the computation independent of the implementation of stateful computation.

* In the name of not getting bogged down in details, there are a few parts of this discussion that are not entirely accurate, particularly around functions like unsafePerformIO[6].

[0] http://hackage.haskell.org/package/base-4.12.0.0/docs/src/GH...

[1] http://hackage.haskell.org/package/base-4.12.0.0/docs/src/GH...

[2] http://hackage.haskell.org/package/base-4.12.0.0/docs/src/GH...

[3] https://acm.wustl.edu/functional/state-monad.php

[4] http://hackage.haskell.org/package/base-4.12.0.0/docs/Contro...

[5] http://hackage.haskell.org/package/mtl-2.2.2/docs/Control-Mo...

[6] http://hackage.haskell.org/package/base-4.12.0.0/docs/System...