The polynomial $2z^3$ is analytic everywhere. Write $z=x+iy$; the additional term is $b(x^2+y^2)^{3/2}$, which depends on both $z$ and its complex conjugate.
For nonzero b its derivative with respect to $\bar z$ is nonzero away from the origin, so it cannot be analytic on an open region. Taking **$b=0$** removes that term and leaves the entire polynomial $2z^3$.