Step 1: Geometric view
The condition says z is equidistant from $5i$ (the point $(0,5)$) and $-3$ (the point $(-3,0)$).
Step 2: Perpendicular bisector
Midpoint $= (-1.5, 2.5)$. The slope of the joining line is $\dfrac{5}{3}$, so the bisector has slope $-\dfrac{3}{5}$.
Its equation is $y-2.5=-\dfrac{3}{5}(x+1.5)$, i.e. $3x+5y=8$.
Step 3: Check the point
$(-1.5,2.5)$: $3(-1.5)+5(2.5) = 8$, which satisfies the equation.
Final Answer:
The perpendicular bisector is 3x + 5y - 8 = 0.
\[ \boxed{\text{(A) }3x+5y-8=0} \]