Imaginary numbers and negative zero
Yes, there is a usage for the negative imaginary zero. But first, I will say something about the negative zero in general.

Why have a negative zero?

First of all, the main reason to have a signed zero for floating points is that floating points have only limited precision and we often want to distinguish the following cases of a real number $x$, represented by a floating point:


$x\geq 0$, which we will denote by $+0$. 
$x&lt;0$, which we will denoted by  $-0$.


As we cannot distinguish between these cases ordinarily in our floating point representation when $|x|$ is smaller than the floating point precision, we need a special notation for this. We want to distinguish between these two almost $0$ numbers, as this difference can be 'enlarged' with the appropriate function: when we have defined $1/0=\infty$ (we work with the extended reals) we have $+0={-0}$, but $1/{-0}={-\infty}\neq +\infty = 1/{+0}$.

The negative imaginary zero

For the following, consider a complex number $z=x+i\cdot y$, with $x$ the real part and $y$ the imaginary part, represented by a floating point for $x$ and another floating point for $y$. 

One important application where we want to distinguish between $-0$ and $+0$ is when computing the inverse elementary functions (i.e. $\ln(z), \sqrt{z}, \arcsin(z)$) for the complex numbers. All these functions are defined in terms of $\ln(z)$, the complex logarithm. 

I won't go into the details why, but this function is discontinuous on the whole negative real number line: $\ln(x + i\cdot +0)\neq \ln(x+i\cdot{-0})$ for all $x\leq 0$, this is half-line known as the branch cut of the complex logarithm. (Note that we also want to know the sign of $x$ here to see whether are 'on' this line in the first place!)

For example, we have $-4+i0 = -4 - i0$, but $2i=\sqrt{-4+i0 }\neq \sqrt{-4-i0}=-2i$. So, we need to distinguishing between $+i0$ and $-i0$ to know on which 'side' of the branch cut we are to compute the complex logarithm $\ln(x)$ and all other inverses of elementary functions defined from it. For more on this topic, see these notes by Prof. W. Kahan.