Given two (radical, prime) ideals $I_1$ and $I_2$ of $\mathcal{O}_{\mathbb{C}^2,0}$ generated by holomorphic functions in two variables $g_1(z_1,z_2)$ and $g_2(z_1,z_2)$ (resp.), one can also consider the ideals $$ I_i^\mathbb{R}=(\mathcal{Re}\ g_i, \mathcal{Im}\ g_i)\subset \mathcal{O}_{\mathbb{R}^4,0}\ \ \ \ , i=1,2.$$
Question: $\psi^*I_1^\mathbb{R}=I_2^\mathbb{R}$ for some smooth diffeomorphism of $\mathbb{R}^4$ (real analytic if necessary) implies that $\phi^*I_1=I_2$ for some complex isomorphism $\phi$ of $\mathbb{C}^2$.
My vague idea: I tried with $$ \mathcal{Re}\ g_1= \mathcal{Re}\ g_2\ h_{RR} + \mathcal{Im}\ g_2 \ h_{RI}$$ $$\mathcal{Im}\ g_1 = \mathcal{Re}\ g_2\ h_{IR} + \mathcal{Im}\ g_2 \ h_{II}$$ and use Cauchy Riemann equations to solve this.