4. Let $m, n$ be non-zero real numbers, $\mathrm{i}$ be the imaginary unit, $z \in \mathbf{C}$, then the graphs of the equations $|z+n \mathrm{i}|+|z-m \mathrm{i}|=n$ and $|z+n \mathrm{i}|-|z-m \mathrm{i}|=-m$ in the same complex plane (where $F_{1}, F_{2}$ are foci) are
According to the problem, for the ellipse, $2 a=n>0, 2 c=|-n-m|=n+m>0$
$\Rightarrow$ the hyperbola is the branch closer to point $F_{2}$, so the answer is $B$.