Since $F$ is bicomplex-holomorphic, we necessarily have $\frac{\partial F}{\partial Z^\dagger} = 0$, and your condition $$ \frac{\partial F}{\partial Z^\dagger}=\mu(Z)\frac{\partial F}{\partial Z} $$ implies that either $\mu$ is the zero measure or $F$ is constant.
By the way, any bicomplex-holomorphic function $F(Z)$ becomes - after a linear change of coordinates - of the form $f_1(w_1) \oplus f_2(w_2)$ for some $\mathbb{C}$-holomorphic functions $f_1$ and $f_2$.