Plot3D doesn't generate the ellipitic paraboloid it's supposed to

Clear["Global`*"]

The default PlotRange is obscuring the detail that you want.

Plot3D[(x - 1)^4 + (2 x + y)^2, {x, -8, 8}, {y, -8, 8},
 PlotRange -> {0, 6},
 ClippingStyle -> None]

ContourPlot[(x - 1)^4 + (2 x + y)^2,
 {x, -8, 8}, {y, -8, 8},
 PlotRange -> {Full, Full, {0, 20}}]

To understand why the plot you made (which is correct although distorted) looks like it does and also see the region that really interests you without distortion, make use of the options AspectRatio and Contours. Like so:

ContourPlot[(x - 1)^4 + (2 x + y)^2, {x, -2, 4}, {y, -14, 9},
  AspectRatio -> Automatic,
  Contours -> 2^Range[-2, 6],
  ImageSize -> Large]

Observe that the countours in the range around y = 0 become more and more vertical as z increases. That explains your result.