By performing some computations using the Singular software, I've noticed the following pattern: if $\mu$ is the Milnor Number of a holomorphic germ $f\in \mathcal{O}_n$ at the origin, then the Milnor Number of its real and imaginary parts are equal and are $\mu^2$, that is, $$ \mu_{(\mathcal{R}e(f),0)} = \mu_{f,0}^2 $$

By Milnor number of the real part of $f$, I mean $u=\mathcal{R}e(f)$ as a germ of real analytic function of $2n$ real variables (the real and imaginary parts of each complex variable). If $\mathcal{A_{2n}}$ is the ring of germs of such real analytic functions, then $$ \mu_{(\mathcal{R}e(f),0)}=\text{dim}_{\mathbb{R}} \dfrac{\mathcal{A}_{2n}}{\left<\frac{\partial u}{\partial x_1},..., \frac{\partial u}{\partial x_n}, \frac{\partial u}{\partial y_1},..., \frac{\partial u}{\partial y_n}\right> } $$

I would like to know if there's any generalisation to this. I've tried using some direct sum properties on ideals but got nowhere. I suspect there might be some tensor products involved, but also got nowhere.

By performing some computations using the Singular software, I've noticed the following pattern: if $\mu$ is the Milnor Number of a holomorphic germ $f\in \mathcal{O}_n$ at the origin, then the Milnor number of its real and imaginary parts are equal and are $\mu^2$, that is, $$ \mu_{(\mathcal{R}e(f),0)} = \mu_{f,0}^2 $$

By Milnor number of the real part of $f$, I mean $u=\mathcal{R}e(f)$ as a germ of real analytic function of $2n$ real variables (the real and imaginary parts of each complex variable). If $\mathcal{A_{2n}}$ is the ring of germs of such real analytic functions, then $$ \mu_{(\mathcal{R}e(f),0)}=\text{dim}_{\mathbb{R}} \dfrac{\mathcal{A}_{2n}}{\left<\frac{\partial u}{\partial x_1},..., \frac{\partial u}{\partial x_n}, \frac{\partial u}{\partial y_1},..., \frac{\partial u}{\partial y_n}\right> } $$

I would like to know if there's any generalisation to this. I've tried using some direct sum properties on ideals but got nowhere. I suspect there might be some tensor products involved, but also got nowhere.