Summary: In this work we investigate \(n\)-widths of multiplier operators \(\Lambda_\ast\) and \(\Lambda\), defined for functions on the complex sphere \(\Omega_d\) of \(\mathbb{C}^d\), associated with sequences of multipliers of the type \(\{\lambda_{m,n}^\ast\}_{m,n\in\mathbb{N}}\), \(\lambda_{m,n}^\ast=\lambda(m+n)\) and \(\{\lambda_{m,n}\}_{m,n\in \mathbb{N}},\lambda_{m,n}=\lambda(\max\{m,n\})\), respectively, for a bounded function \(\lambda\) defined on \([0,\infty)\). If the operators \(\Lambda_\ast\) and \(\Lambda\) are bounded from \(L^p(\Omega_d)\) into \(L^q(\Omega_d)\), \(1\leq p\), \(q\leq\infty\), and \(U_p\) is the closed unit ball of \(L^p(\Omega_d)\), we study lower and upper estimates for the \(n\)-widths of Kolmogorov, linear, of Gelfand and of Bernstein, of the sets \(\Lambda_\ast U_p\) and \(\Lambda U_p\) in \(L^q(\Omega_d)\). As application we obtain, in particular, estimates for the Kolmogorov \(n\)-width of classes of Sobolev, of finitely differentiable, infinitely differentiable and analytic functions on the complex sphere, in \(L^q(\Omega_d)\), which are order sharp in various important situations.