Clear[α, f, x, y, U, F, A, B, h, T, G, 
        NF, r, s, t]
\(z_\&_ := 
      z /. Complex[a_, b_] :> 
          Complex[a, \(-b\)]\)
\[LeftDoubleBracketingBar]z_\[RightDoubleBracketingBar] := 
        Simplify[(Sqrt[z Conjugate[z]])]

i = 
        1;
α =  i + 
              1 - (Sqrt[(((i + 1))^2) - 1]) // N // 
          Chop;
f[λ_] = (((λ^2) - (α^2))/(1 - (α^2) (λ^2)));
U[λ_, x_, 
          y_] = ((((λ^(-1)) (x + I y) - λ (x - I y)))/4);
F[λ_, x_, y_]=(1/2)(({{(E^U)[λ, x, y] + (E^(-U[λ, x, y])), (E^U)[λ, x, y] - (E^(-U[λ, x, y]))}, {(E^U)[λ, x, y] - (E^(-U[λ, x, y])), (E^U)[λ, x, y] + (E^(-U[λ, x, y]))}}));

A[λ_, x_, 
          y_] = F[λ, x, y][[1, 1]];
B[λ_, x_, y_] = F[λ, x, y][[1, 2]];

sig3=(({{1, 0}, {0, -1}}));
(Subscript[π, 1])=(({{1, 0}, {0, 0}}));
(Subscript[π, 2])=(({{0, 0}, {0, 1}}));
\(π\&~[λ_, x_, y_]\)=1/(\[LeftDoubleBracketingBar]A[α, x, 
                      y]\[RightDoubleBracketingBar]^2 + \[LeftDoubleBracketingBar]α(\[RightDoubleBracketingBar]^(-2)) \[LeftDoubleBracketingBar]B[α, x, y](\[RightDoubleBracketingBar]^2)) (({{\[LeftDoubleBracketingBar]A[α, x, 
                          y](\[RightDoubleBracketingBar]^2), \(λ\ \(α\^\(-1\)\) \(\(A[α, x, y]\)\&_\) B[α, x, y]\)}, {\(λ\^\(-1\)\ \(\(α\^\(-1\)\)\&_\) 
                    A[α, x, 
                      y] \(B[α, x, y]\)\&_\), \[LeftDoubleBracketingBar]α(\[RightDoubleBracketingBar]^(-2)) \[LeftDoubleBracketingBar]B[α, x, y](\[RightDoubleBracketingBar]^2)}}));
\(\(π\^*\)[λ_, x_, y_]\)=1/(\[LeftDoubleBracketingBar]A[α, x, 
                      y]\[RightDoubleBracketingBar]^2 + \[LeftDoubleBracketingBar]α(\[RightDoubleBracketingBar]^(-2)) \[LeftDoubleBracketingBar]B[α, x, y](\[RightDoubleBracketingBar]^2)) (({{ \[LeftDoubleBracketingBar]α(\[RightDoubleBracketingBar]^(-2)) \[LeftDoubleBracketingBar]B[α, x, y](\[RightDoubleBracketingBar]^2), \(\(-λ\)\ \(α\^\(-1\)\) \(\(A[α, x, y]\)\&_\) B[α, x, y]\)}, {\(\(-λ\^\(-1\)\)\ \(\(α\^\(-1\)\)\&_\) 
                    A[α, x, 
                      y] \(B[α, x, y]\)\&_\), \[LeftDoubleBracketingBar]A[α, x, y](\[RightDoubleBracketingBar]^2)}}));
h[λ_] = (1/(Sqrt[f]))[λ] (Subscript[π, 1]) + (Sqrt[f])[λ] (Subscript[π, 2]);
\(h\&~[λ_, x_, 
          y_] = \(\@f[λ]\) π\&~[λ, x, 
              y] + \(1\/\@f[λ]\) \(π\^*\)[λ, x, 
              y];\)
\(T[x_, 
            y_] = \((\(\(A[α, x, y]\)\&_\) 
                B[α, x, 
                  y] \((1 + α\^2)\))\)/\((α \[LeftDoubleBracketingBar]A[α, x, 
                        y]\[RightDoubleBracketingBar]^2 - \ \(α\&_\) \[LeftDoubleBracketingBar]B[α, x, 
                        y]\[RightDoubleBracketingBar]^2)\);\)

G[λ_, x_, y_]=(I/(Sqrt[\[LeftDoubleBracketingBar]T[x, y](\[RightDoubleBracketingBar]^(2 )) + 1]))(({{-1, -λ T[x, y]}, {\(\(-λ\^\(-1\)\)\ \(T[x, y]\)\&_\), 1}}));
\(NF[λ_, x_, y_] = 
        h[λ] . F[λ, x, y] . h\&~[λ, x, y] . 
          G[λ, x, y];\)
\(\(NF\^*\)[x_, y_] = 
        D[NF[λ, x, y], λ] /. λ -> 
            1;\)
\(f[x_, 
          y_] = \(-1\)/2 
             NF[1, x, y] . \((I\ sig3)\) . 
              Inverse[NF[1, x, y]] - \((I\ \(NF\^*\)[x, y]\ )\) . 
            Inverse[NF[1, x, y]];\)
r[x_, y_] = 
        Chop[N[I (Part[f[x, y], 1, 2] + 
                  Part[f[x, y], 2, 1])]];
s[x_, y_] = 
        Chop[N[Part[f[x, y], 1, 2] - Part[f[x, y], 2, 1]]];
t[x_, y_] = 
          Chop[N[I (- Part[f[x, y], 1, 1] + 
                    Part[f[x, y], 2, 2])]];

Bubble = 
      ParametricPlot3D[{r[x, y] // N // Chop, s[x, y] // N // 
            Chop, t[x, y] // N // Chop}, {x, 0, 4}, {y, 0, 2  Pi}, 
        PlotPoints -> {26, 51}, PlotRange -> All, 
        AxesLabel -> {r, s, t}]

\( (*Parallel\ surface*) 
\(ff[x_, y_] = 
        1/2  NF[1, x, y] . \((I\ sig3)\) . 
              Inverse[NF[1, x, y]] - \((I\ \(NF\^*\)[x, y]\ )\) . 
            Inverse[NF[1, x, y]];\)\)
rr[x_, y_] = 
        Chop[N[I (Part[ff[x, y], 1, 2] + 
                  Part[ff[x, y], 2, 1])]];
ss[x_, y_] = 
        Chop[N[Part[ff[x, y], 1, 2] - Part[ff[x, y], 2, 1]]];
tt[x_, y_] = 
        Chop[N[I (- Part[ff[x, y], 1, 1] + 
                  Part[ff[x, y], 2, 2])]];

ParaBubble = 
      ParametricPlot3D[{rr[x, y] // N // Chop, ss[x, y] // N // 
            Chop, tt[x, y] // N // Chop}, {x, 0, 4}, {y, 0, 2  Pi}, 
        PlotPoints -> {26, 51}, PlotRange -> All, 
        AxesLabel -> {r, s, t}]