Is there a context where delta(x+i*c), where c is a real number, makes sense?
It came up while I was doing an Inverse Fourier Transform, and I failed to appreciate its significance.
Does anyone know of a situation where it's normal to see things like delta(z)?
Dirac delta is really not a function. It cannot be precisely defined like that even for real numbers. Most of its properties are derived from how it acts on a specific function, most notably
$$ \int_{-\infty}^{+\infty}f(x)\delta(x) dx = f(0) $$
or more precisely within Lebesgue
$$ \int_{-\infty}^{+\infty}f(x) \delta \{ dx \} = f(0) $$
Is there anything like that within a complex domain out of the box? Sure. Yet in this case, Dirac delta is actually defined without resorting to any sort of approximations, it comes very much directly from Cauchy's integral formula.
$$ f(z) = \frac{1}{2\pi i} \oint_{\partial H} \frac{f(s)}{s-z}\,ds,\quad z\in H $$
Different from real domain, as long as $f(z)$ is behaving nicely within a domain $H$, you may not have to squeeze $H$ after all where within real domain you have to take some sort of limiting process.
$$ \delta[f] = \frac{1}{2\pi i} \oint_{\partial H} \frac{f(s)}{s-z}\,ds$$
Notice that $z$ is fixed, although any point within $H$.
Since this formula is giving you a hint about the complex delta, we can go back and derive complex approximations for the complex Dirac delta, yet we did this because we had to, regarding reals. We do not have to do it for complex. (Notice, however, that, for example, we do not have infinity defined the same way we have for reals.)