With the help of the answer of the question Amelioration of a 3D plot, I have been able to construct the following plot

SetOptions[{Plot, ListPlot, ArrayPlot, ContourPlot, DiscretePlot3D,     Plot3D, Graph3D},    BaseStyle -> {14, Directive[FontFamily -> "Times New Roman"]}]; arrowAxes[arrowLength_] :=   Map[{Black, Arrow[Tube[{{0, 0, 0}, #}]]} &,    arrowLength IdentityMatrix[3]] With[{zproj = -1.5},   Module[{f, surface, graphicsComplexPts, contourParts, contourProjPts,     projection}, f[x_, y_] := -(x^2 + y^2)^.2;   surface =     Plot3D[f[x, y], {x, -1, 1}, {y, -1, 1}, PlotRange -> {-1, 0},      ClippingStyle -> None, MeshFunctions -> {#3 &}, Mesh -> 15,      MeshStyle -> {Opacity[.5], Red, Thickness[.0075]}, Boxed -> False,      Axes -> False, PlotPoints -> 200,      PlotStyle -> {Opacity[0.5], Cyan}, Lighting -> "Neutral"];   (*Plot3D returns a Graphics3D object with a GraphicsComplex,   first element of which is a list of points*)   graphicsComplexPts = surface[[1, 1]];   (*Find Line heads within GraphicsComplex*)   contourParts = Position[surface, _Line];   (*using GraphicsComplex indexing,map to projected contours.*)   contourProjPts[k_] :=     graphicsComplexPts[[surface[[Sequence @@ contourParts[[k]],          1]]]] /. {x_, y_, z_} -> {x, y, zproj};   (*make projection with graphics primitives*)   projection =     Graphics3D[{Red,       Table[Line@contourProjPts[k], {k, 1, Length@contourParts}],       arrowAxes[.8]}];   Show[{surface, projection}, PlotRange -> {All, All, {zproj, 0.1}},     BoxRatios -> {1, 1, 1.4}]]] 

But I have not been able to place correctly the origin where I want — on the projected hyperplan —, to have the same color for the background of the level set than for the surface, to put {x, y, z} label on the axes and to downsize the tube and the arrowheads.