Hello Ehund,
it seems you totally solved your problem. That is good.
But another remark. I think it is a very interesting idea to do geometry in R2 with complex numbers. As my Mathematica (Version 7) does not know ComplexRegionPlot I strived to find a workaround and did it like this
toZ[{x_, y_}] := x + I y (* point to complex number*)
frZ[z_] := {Re[z], Im[z]} (* complexnumber to point *)
trZ[z_, a_] := z + a (* translation *)
roZ[z_, a_] := z Exp[I a] (* rotation *)
Here we have a semicircle
RegionPlot[
And[Norm[{x, y} - {0, 0}] < 1, {x, y}[[2]] > 0],
{x, -1.5, 1.5}, {y, -1.5, 1.5}] (*seimicircle *)
And here is the rotated and translated semicircle
tt = -.6 + .5 I;
ww = -1.2;
RegionPlot[
And[
Norm[frZ[roZ[trZ[toZ[{x, y}], tt], ww]]] < 1,
frZ[roZ[trZ[toZ[{x, y}], tt], ww]][[2]] > 0],
{x, -2, 2}, {y, -2, 2}]