Code:

rpic = ParametricPlot3D[{{Re[z], Im[z], Re[Sqrt[z]]}, {Re[z], Im[z],
Re[-Sqrt[z]]}} /. z -> r Exp[I t], {r, 0, 2}, {t, -\[Pi], \[Pi]}]
ipic = ParametricPlot3D[{{Re[z], Im[z], Im[Sqrt[z]]}, {Re[z], Im[z],
Im[-Sqrt[z]]}} /. z -> r Exp[I t], {r, 0, 2}, {t, -\[Pi], \[Pi]}]
realpic = Show[rpic, PlotLabel -> Style["Real", 16]]
imagpic = Show[ipic, PlotLabel -> Style["Imag", 16]]
rprincp =
ParametricPlot3D[{Re[z], Im[z], Re[Sqrt[z]]} /. z -> r Exp[I t], {r,
0, 2}, {t, -\[Pi], \[Pi]},
PlotLabel -> Style["Real principal branch", 16]]
iprincp =
ParametricPlot3D[{Re[z], Im[z], Im[Sqrt[z]]} /. z -> r Exp[I t], {r,
0, 2}, {t, -\[Pi], \[Pi]},
PlotLabel -> Style["Imag principal branch", 16]]
GraphicsGrid[{{realpic, imagpic}, {rprincp, iprincp}}]