The "Paraboloidoid"

The parabola over the complex numbers :  u+vi=(((a+bi))^2)
Constraints:  u=(a^2)-(b^2),  v= 2 a b

Equations

{(a,b,v): 2ab-v}
{(a,b,u): (a^2)-(b^2)-u} 
{(b,u,v):4 (b^4)+4 (b^2) u-(v^2)}
{(a,u,v):4 (a^4)-4 (a^2) u-(v^2)}

Projection onto (a,b,u)

ContourPlot3D[a^2-b^2 -u==0,{a,-5,5},{b,-5,5},{u,-5,5},
ColorFunction->Function[{a,b,u},ColorData["BlueGreenYellow"][2a  b]]]

Projection onto (a,b,v)

ContourPlot3D[2a b-v==0,{a,-5,5},{b,-5,5},{v,-5,5},
ColorFunction->Function[{a,b,v},ColorData["BlueGreenYellow"][a^2-b^2]]]

Projection onto (a,u,v)

ContourPlot3D[-4a ^2 u+4a^4-v^2==0,{a,-5,5},{u,-5,5},{v, -5,5},
ColorFunction->Function[{a,u,v},ColorData["BlueGreenYellow"][v/2a]]]

Projection onto (b,u,v)

ContourPlot3D[4b^2 u+4b^4-v^2==0,{b,-5,5},{u,-5,5},{v,-5,5}, 
ColorFunction->Function[{b,u,v},ColorData["BlueGreenYellow"][v/2b]]]