Let $X$ be an infinite set and let $\text{End}(X)$ be the set of all functions $f:X\to X$. For $f\in\text{End}(X)$ let $$\text{Com}(f) = \{g\in\text{End}(X): g\circ f = f \circ g\}.$$ Is there $f\in \text{End}(X)$ such that $\text{Com}(f) = \{\text{id}_X, f\}$?
If not, what is $\min\{|\text{Com}(f)|:f\in\text{End}(X)\}$, in terms of $|X|$?