Simplify[{Re[#]==0,Im[#]==0}&@Expand[((((x+ I y))^2)-(((z+ I w))^3)-(((z+ I w))^2))], Assumptions->{x∈Reals, y∈Reals,z∈Reals,w∈Reals}]

{#[[1,2]],#[[2,2]],z}&/@Solve[{(w^2)+(x^2)+3 (w^2) z==(y^2)+(z^2)+(z^3),(w^3)+2 x y==w z (2+3 z)},{x,y}]

frame[w_]:=ParametricPlot3D[
{{-((Sqrt[-(w^2)-3 (w^2) z+(z^2)+(z^3)+(Sqrt[((((w^2)+(z^2)))^2) (1+(w^2)+2 z+(z^2))])])/(Sqrt[2])),(1/((w^3)-2 w z-3 w (z^2)))((Sqrt[2]) (w^2) (Sqrt[-(w^2)-3 (w^2) z+(z^2)+(z^3)+(Sqrt[((((w^2)+(z^2)))^2) (1+(w^2)+2 z+(z^2))])])+3 (Sqrt[2]) (w^2) z (Sqrt[-(w^2)-3 (w^2) z+(z^2)+(z^3)+(Sqrt[((((w^2)+(z^2)))^2) (1+(w^2)+2 z+(z^2))])])-(Sqrt[2]) (z^2) (Sqrt[-(w^2)-3 (w^2) z+(z^2)+(z^3)+(Sqrt[((((w^2)+(z^2)))^2) (1+(w^2)+2 z+(z^2))])])-(Sqrt[2]) (z^3) (Sqrt[-(w^2)-3 (w^2) z+(z^2)+(z^3)+(Sqrt[((((w^2)+(z^2)))^2) (1+(w^2)+2 z+(z^2))])])+((((-(w^2)-3 (w^2) z+(z^2)+(z^3)+(Sqrt[((((w^2)+(z^2)))^2) (1+(w^2)+2 z+(z^2))])))^(3/2))/(Sqrt[2]))),z},{((Sqrt[-(w^2)-3 (w^2) z+(z^2)+(z^3)+(Sqrt[((((w^2)+(z^2)))^2) (1+(w^2)+2 z+(z^2))])])/(Sqrt[2])),(1/((w^3)-2 w z-3 w (z^2)))(-(Sqrt[2]) (w^2) (Sqrt[-(w^2)-3 (w^2) z+(z^2)+(z^3)+(Sqrt[((((w^2)+(z^2)))^2) (1+(w^2)+2 z+(z^2))])])-3 (Sqrt[2]) (w^2) z (Sqrt[-(w^2)-3 (w^2) z+(z^2)+(z^3)+(Sqrt[((((w^2)+(z^2)))^2) (1+(w^2)+2 z+(z^2))])])+(Sqrt[2]) (z^2) (Sqrt[-(w^2)-3 (w^2) z+(z^2)+(z^3)+(Sqrt[((((w^2)+(z^2)))^2) (1+(w^2)+2 z+(z^2))])])+(Sqrt[2]) (z^3) (Sqrt[-(w^2)-3 (w^2) z+(z^2)+(z^3)+(Sqrt[((((w^2)+(z^2)))^2) (1+(w^2)+2 z+(z^2))])])-((((-(w^2)-3 (w^2) z+(z^2)+(z^3)+(Sqrt[((((w^2)+(z^2)))^2) (1+(w^2)+2 z+(z^2))])))^(3/2))/(Sqrt[2]))),z}}
,{z,-4,4}
, PlotRange->4, PlotStyle->Directive[Blue, Thick], Boxed->False, AxesOrigin->{0,0,0}(*, RegionFunction->Function[{x,y,z},!(Abs[x]<0.08&&z>0)]*)]

frame[-1]

Rasterize[frame[-0.01], ImageResolution->250]

Export["T:\\BGabor\\algebraic2.avi", Table[Rasterize[frame[w], ImageResolution->300], {w,-4+0.001,4,0.01}]]