= Solution
Fix the measure conventions before evaluating this <zero-dimensional supersymmetric field theory>. Take $dz\,d\bar z$ to mean ordinary positive area measure $d^2z=dx\,dy$, and choose <Berezin integration> orientations with $\int d^2\theta\,\theta_1\theta_2=\int d^2\overline\theta\,\overline\theta_1\overline\theta_2=1$. A different overall measure normalization multiplies all the unnormalized answers by the same constant. Assume $r=\deg P=\deg W-1\ge1$, so $|P(z)|\to\infty$ at infinity and all polynomial insertions converge. An affine or constant $W$ does not give this confining integral and is outside this convergence hypothesis.
The <Grassmann variables> truncate the fermionic exponential. The coefficient of $\theta_1\theta_2\overline\theta_1\overline\theta_2$ is $P'(z)\overline{P'(z)}$, so the <partition function> becomes
$$
Z=\int_{\mathbb C}d^2z\,|P'(z)|^2e^{-|P(z)|^2}.
$$
The <Jacobian determinant> of the <holomorphic map> $w=P(z)$ is $|P'(z)|^2$. Away from its finitely many critical values, a polynomial of <polynomial degree> $r$ has $r$ preimages counted with multiplicity. The <change of variables formula> therefore gives
$$
\boxed{Z=r\int_{\mathbb C}d^2w\,e^{-|w|^2}=\pi r=\pi(\deg W-1).}
$$
Critical values form a set of area zero and do not affect the integral. This <partition function> measures the covering degree, even when some zeros of $P$ are repeated.
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