Liouville function and perfect squares for Gaussian integers

Question

Let $\lambda$ be the Liouville function. It is a well known fact that the characteristic function of the perfect squares, denoted by $\chi_2$, can be written as $\chi_2(n)=\sum_{d|n}\lambda(d)$. My question is there an analogous for the characteristic function of perfect Gaussian squares and the Liouville function for Gaussian integers?