Let $A$ be a finite dimensional commutative algebra. We can assume that it is local.

Question: Which such $A$ have the property that every finite dimensional $A$-module has complexity at most 1? (This should be equivalent to the simple module having complexity equal to one or equivalently bounded Betti numbers)

Note that a module has complexity at most one if and only if the terms in a minimal projective resolution have bounded dimensions.

Examples include $A=K[x]/(x^n)$.