Looks like the "stuff" outside an ellipse.
First find |z - 4 - i| + |z + 4 + i| = 1 to get the boundary.
Note that this can be written as |z - (4 + i)| + |z - (-4 - i)| = 1.
Now ..... if you're familiar with the locus definition of an ellipse then there's a very simple way of seeing that there's no solution to this equation.
Therefore the answer to the inequality is the entire complex plane ....
In your (unnecessary) re-posting of this question you said:
So what's it meant to be then, an inequality or an equality? Either way, my first post contains the answer.
If you need an algebraic approach where you substitute z = x + iy etc. please say so.