Let \(H\) be a separable complex Hilbert space. A pair \((A,B)\) of commuting operators on \(H\) is called hypercyclic if there exists an \(x\in H\) such that \(\{A^nB^k\mid n,k\geq 0\}\) is dense in \(H\). The author studies hypercyclicity properties of pairs \((M_f^{*}, M_g^{*})\), where \((f,g)\in H^{\infty}(G)\) are bounded analytic functions on an open subset \(G\) of \(\mathbb{C}\), and \(M_f\) and \(M_g\) are the associated multiplication operators. When \(G\) has finitely many components, it is shown that \((M_f^{*}, M_g^{*})\) is hypercyclic if and only if the semigroup generated by \(M_f\) and \(M_g\) contains a hypercyclic operator (i.e., an operator with a dense orbit). This result is not true anymore when \(G\) has infinitely many components.