= Solution
First there is a necessary coordinate qualification. With the literal $z$ action obtained in part (c), the temporal weights are $-1,+1$ and reflection is conjugation followed by exchange. A linear term with $(\mu+i\omega)$ in both equations fails reflection equivariance for $\omega\ne0$, since conjugation changes the sign of $i\omega$. The proposed first-component <monomial> $\bar z_1z_2^2$ also has temporal weight $+3$, rather than $-1$. Thus the displayed equations cannot be proved in those literal coordinates. They are the correct <dihedral fourfold Hopf normal form> in the common-phase Hopf coordinates $(q_1,q_2)=(z_1,\bar z_2)$.
In these coordinates the temporal circle imposes total phase weight one on every vector-field <monomial>. There are no quadratic <monomials> with that weight. A cubic <monomial> has two unbarred factors and one barred factor. For the first component, rotation by $\rho$ must multiply the <monomial> by $i$. Inspecting the six phase-compatible <monomials> leaves precisely
$$
q_1^2\bar q_1=q_1|q_1|^2,\qquad q_1q_2\bar q_2=q_1|q_2|^2,\qquad q_2^2\bar q_1.
$$
The excluded three <monomials> multiply by $-i$ instead. Reflection swaps the components, so their coefficients must be the same after swapping indices. The linear part is a scalar complex multiple by absolute irreducibility. Orient the Hopf parameter to write it as $\mu+i\omega$, and name the cubic coefficients $-a,-b,-c$, where $a,b,c$ are generally complex. This proves
$$
\boxed{\begin{aligned}
\dot q_1&=(\mu+i\omega)q_1-aq_1|q_1|^2-bq_1|q_2|^2-c\bar q_1q_2^2,\\
\dot q_2&=(\mu+i\omega)q_2-aq_2|q_2|^2-bq_2|q_1|^2-c\bar q_2q_1^2.
\end{aligned}}
$$
These are cubic amplitude <normal forms of a dynamical system>, after the usual symmetry-compatible Hopf normalization. Returning to literal $z$ coordinates conjugates the second equation: its linear coefficient is $\mu-i\omega$, its cubic coefficients are $\bar a,\bar b,\bar c$, and the first coupling is $-c\bar z_1\bar z_2^2$. That corrected form respects every printed generator.
On the three <complex axial isotropy subgroup> fixed spaces, substitute $q_1=Re^{i\Omega t}$ and $q_2=0,q_1,iq_1$. The radial equations are $\dot R=R(\mu-A_jR^2)$, with
$$
A_R=\operatorname{Re}a,\qquad A_A=\operatorname{Re}(a+b+c),\qquad
A_D=\operatorname{Re}(a+b-c).
$$
Therefore the leading component amplitudes are
$$
\boxed{R_R^2=\frac\mu{\operatorname{Re}a},\qquad
R_A^2=\frac\mu{\operatorname{Re}(a+b+c)},\qquad
R_D^2=\frac\mu{\operatorname{Re}(a+b-c)}.}
$$
Each expression applies on the parameter side where it is positive, with its denominator nonzero for a nondegenerate branch. The frequencies are $\Omega_j=\omega-\operatorname{Im}(a_j)R_j^2$, with $a_j=a,a+b+c,a+b-c$. The denominators are generically distinct, so coexisting branches generically have different component amplitudes. If the amplitude is instead measured by the original real-space norm, the identity $|w_1|^2+|w_2|^2=2(|q_1|^2+|q_2|^2)$ gives squared amplitudes $2R_R^2$, $4R_A^2$, and $4R_D^2$, again generically distinct. Equalities require special coefficient relations, rather than symmetry forcing them.
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