= Solution
{c}
The <continuously differentiable function> $z\bar z=x^2+y^2$ is a particular solution, since
$$
\frac{\partial}{\partial\bar z}(x^2+y^2)
=\frac12(2x+i,2y)=z.
$$
If $f$ is any solution, set $h(z)=f(z)-z\bar z$. Then $h$ is <continuously differentiable> and its <Wirtinger derivative> vanishes. By the preceding part and the sufficient <Cauchy-Riemann equations> criterion, $h$ is a <holomorphic function> on all of $\mathbb C$, hence an <entire function>.
Conversely, every <entire function> $h$ has vanishing <Wirtinger derivative>, so addition of $h$ preserves the prescribed derivative. Therefore all solutions are
$$
\boxed{f(z)=|z|^2+h(z),\qquad h\text{ any entire function}.}
$$
This is an <inhomogeneous Wirtinger equation with holomorphic forcing>: a particular solution plus an arbitrary element of the homogeneous solution space.
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