The problem is the increment operator now gives two possible answers to a++. This can't be done without relying on some notion of state. That state has to be stored somewhere.
The distinction you're looking for is "does this operator store the information required for idempotence in the output value, or is it stored elsewhere?" If it's stored in the output value, good. If it's stored elsewhere the operator is not a pure function.
In which case your complain doesn't actually have anything to do with idempotence. Your complaint is that the function which is claiming to be idempotent is not even a function.
You can't claim your code has the property f(f(x)) = f(x), if your code doesn't even have the property f(x) = f(x).
I'm not sure operators are a good example, they can be idempotent or not depending on the implementation. The operator is a function that takes two values and return a result: result = op(left, right).
The distinction you're looking for is "does this operator store the information required for idempotence in the output value, or is it stored elsewhere?" If it's stored in the output value, good. If it's stored elsewhere the operator is not a pure function.
In which case your complain doesn't actually have anything to do with idempotence. Your complaint is that the function which is claiming to be idempotent is not even a function.
You can't claim your code has the property f(f(x)) = f(x), if your code doesn't even have the property f(x) = f(x).