Distance between vertex and focus is $a(e-1)=4$
$=> a=4$
$\therefore b^2=a^2(e^2-1)=48$
Distance of vertex from center is a.
Coordinate of the center is $(-2,3)$
Equation of the required hyperbola
$\large\frac{(x+2)^2}{16} - \frac{(y-3)^2}{48}=1$
Hence C is the correct answer.