Can the poles of a complex function $f(z)$ be defined as the locations where $\lvert f(z) \rvert = \infty$? According to Wikipedia,

A zero of a meromorphic function $f$ is a complex number $z$ such that $f(z) = 0$. A pole of $f$ is a zero of $1/f$.

Is there a reason why a pole cannot be defined as the location where $\lvert f(z) \rvert = \infty$? I was imagining this to be the case based on this plot of the magnitude of the complex gamma function:


I'd say that your assertion is approximately correct, since one characterization of a pole of $f$ at $z_o$ (as opposed to essential singularity) is that $\lim_{z\to z_o} |f(z)|=+\infty$. In fact, although, there are complications in talking about "the value $\infty$", it can be made to make sense for poles: the function takes values on the Riemann sphere $\mathbb C\mathbb P^1$, which is obtained by adding (in suitable fashion) a "point at infinity" to the complex plane $\mathbb C$.
The behavior of $f$ with isolated singularity that is not a pole, but is an essential singularity, is not only captured by the negation of the condition for a pole, but, further, by the Casorati-Weierstrass theorem: in every neighborhood of an essential singularity $z_o$ of $f$, $f$ comes arbitrarily near to every value in $\mathbb C$.
