Type (3,4,6)

ω1[z_] :=I z;
ω2[z_] := z^2+a;
ω3[z_] := I z^4+I;

f[r_,t_]=ComplexExpand[Re[Integrate[{ω1[z], ω2[z], ω3[z]},z]/.z-> r E^(I t)],TargetFunctions->{Re,Im}]//Simplify

ParametricPlot3D[f[r,t]/{2,5,10}/.a->1/2,
{r,0,2.3},{t,0,2Pi},PlotPoints->{20,100},AxesLabel->{x,y,z},PlotRange->All]

Embedded :

F[x_,y_]=ComplexExpand[Re[Integrate[{ω1[z], ω2[z], ω3[z]},z]/.z-> x+I y],TargetFunctions->{Re,Im}]//Simplify

F[x1,y1]-F[x2,y2]//Factor

Case 1 : x2 ≠ 0

Solve[-x1 y1+x2 y2==0,y2]

F[x1,y1]-F[x2,y2]/.y2->((x1 y1)/x2)//Factor

Eliminate[{ (3 a x2+(x1^2) x2+x1 (x2^2)+(x2^3)+3 x1 (y1^2)), y1 (5 (x2^4)-5 (x1^3) (x2^5)-5 (x1^2) (x2^6)-5 x1 (x2^7)-10 (x1^2) (x2^4) (y1^2)+(x1^4) (y1^4)+(x1^3) x2 (y1^4)+(x1^2) (x2^2) (y1^4)+x1 (x2^3) (y1^4)+(x2^4) (y1^4))}==0,y1]

Factor[(a^3) (x2^3) (27 (x1^4)+27 (x1^3) x2+27 (x1^2) (x2^2)+27 x1 (x2^3)+27 (x2^4))+(a^2) (x2^3) (27 (x1^6)+54 (x1^5) x2+81 (x1^4) (x2^2)+351 (x1^3) (x2^3)+81 (x1^2) (x2^4)+54 x1 (x2^5)+27 (x2^6))+a (x2^3) (9 (x1^8)+27 (x1^7) x2+135 (x1^2) (x2^2)+54 (x1^6) (x2^2)+117 (x1^5) (x2^3)+126 (x1^4) (x2^4)+117 (x1^3) (x2^5)+54 (x1^2) (x2^6)+27 x1 (x2^7)+9 (x2^8))-(x2^3) (-(x1^10)-4 (x1^9) x2-45 (x1^4) (x2^2)-10 (x1^8) (x2^2)-45 (x1^3) (x2^3)-2 (x1^7) (x2^3)-45 (x1^2) (x2^4)+7 (x1^6) (x2^4)+20 (x1^5) (x2^5)+7 (x1^4) (x2^6)-2 (x1^3) (x2^7)-10 (x1^2) (x2^8)-4 x1 (x2^9)-(x2^10))]

This factor is positive for a > 0:

Minimize[3 a+(x1^2)+x1 x2+(x2^2)/.a->0,{x1,x2}]

This factor is nonnegative for a = 1/2 (but not for a = 1!) :

Minimize[(9 (a^2) (x1^4)+6 a (x1^6)+(x1^8)+9 (a^2) (x1^3) x2+12 a (x1^5) x2+3 (x1^7) x2+45 (x1^2) (x2^2)+9 (a^2) (x1^2) (x2^2)+18 a (x1^4) (x2^2)+6 (x1^6) (x2^2)+9 (a^2) x1 (x2^3)+108 a (x1^3) (x2^3)-7 (x1^5) (x2^3)+9 (a^2) (x2^4)+18 a (x1^2) (x2^4)-6 (x1^4) (x2^4)+12 a x1 (x2^5)-7 (x1^3) (x2^5)+6 a (x2^6)+6 (x1^2) (x2^6)+3 x1 (x2^7)+(x2^8))/.a->1/2,{x1,x2}]

All ok for a = 1/2 :

Reduce[((9 (a^2) (x1^4)+6 a (x1^6)+(x1^8)+9 (a^2) (x1^3) x2+12 a (x1^5) x2+3 (x1^7) x2+45 (x1^2) (x2^2)+9 (a^2) (x1^2) (x2^2)+18 a (x1^4) (x2^2)+6 (x1^6) (x2^2)+9 (a^2) x1 (x2^3)+108 a (x1^3) (x2^3)-7 (x1^5) (x2^3)+9 (a^2) (x2^4)+18 a (x1^2) (x2^4)-6 (x1^4) (x2^4)+12 a x1 (x2^5)-7 (x1^3) (x2^5)+6 a (x2^6)+6 (x1^2) (x2^6)+3 x1 (x2^7)+(x2^8))/.a->1/2)==0,{x1,x2},Reals]

Case 2: x2 = 0 :

{-x1 y1+x2 y2,(1/3) (3 a x1+(x1^3)-3 a x2-(x2^3)-3 x1 (y1^2)+3 x2 (y2^2)),(1/5) (-5 y1-5 (x1^4) y1+10 (x1^2) (y1^3)-(y1^5)+5 y2+5 (x2^4) y2-10 (x2^2) (y2^3)+(y2^5))}/.x2->0//Factor

Case : x1 = 0

{-x1 y1,(1/3) x1 (3 a+(x1^2)-3 (y1^2)),(1/5) (-5 y1-5 (x1^4) y1+10 (x1^2) (y1^3)-(y1^5)+5 y2+(y2^5))}/.x1->0//Factor

Minimize[(5+(y1^4)+(y1^3) y2+(y1^2) (y2^2)+y1 (y2^3)+(y2^4)),{y1,y2}]

Case: y1 == 0, x1≠0

{-x1 y1,(1/3) x1 (3 a+(x1^2)-3 (y1^2)),(1/5) (-5 y1-5 (x1^4) y1+10 (x1^2) (y1^3)-(y1^5)+5 y2+(y2^5))}/.y1->0//Factor

Embedded ok!!

Regularity :

Factor[Cross@@Transpose[D[F[x,y],{{x,y}}]]]

%/.x->0//Factor

%/.y->0

Properness:

{-x y,(1/3) x (3 a+(x^2)-3 (y^2)),-(1/5) y (5+5 (x^4)-10 (x^2) (y^2)+(y^4))}

Suppose there is an unbounded  sequence {x,y} such that F[x,y] stays in compact set. Then Either x -> 0 or y -> 0
by the first coordinate.

If y -> 0 then x -> ∞. Assume x y bounded, then xy^2 -> 0, so second coord -> ∞.

Thus x -> 0 and x y and x y^2 bounded and y -> ∞.
    Thus x^2 y^3 bounded, x^4 y -> 0 and thus last coord is unbounded.