Find all solutions for these two equations

To Plot the intersection points of the two contours we can also use the tricks of MeshFunction and Mesh

BTW, I also belive that we can extract the intersection points from the ContourPlot, but I'm unable to find the best way.

Clear["`*"]
fun1 = ((9/10)^x)*Cos[5 x] - y^3;
fun2 = x^2 + 1 x*y^2 + 3*y^4 - 8*y - 4;
a = ContourPlot[fun1 == 0, {x, -5, 5}, {y, -5, 5}, 
   MeshFunctions -> {Function[{x, y}, fun1], Function[{x, y}, fun2]}, 
   Mesh -> {{0}}, MeshStyle -> Directive[PointSize[Large], Red], 
   ContourStyle -> Purple];
b = ContourPlot[fun2 == 0, {x, -5, 5}, {y, -5, 5}, 
   ContourStyle -> Cyan];
Show[b, a]

Try GraphicsMeshFindIntersections

bild = ContourPlot[{fun1 == 0, fun2 == 0}, {x, -5, 5}, {y, -5, 5}];

intersection points of the two curves in bild

s=   Graphics`Mesh`FindIntersections[bild[[All, 1]] 
(*{{-2.81431, 0.437137}, {-2.19896,0.101216}, {-1.57194, -0.197878},
{-0.931073,-0.395926},{-0.333723, -0.464556}, {0.333229, -0.449604}, 
{0.93169, -0.365035},{1.57238, -0.1831}, {1.72927, -0.122129}, 
{2.19959,0.109049}, {2.74194, 0.674778}}*)

Show[{bild, Graphics[Point[sp]]}]

Sorry, don't know why there is an additional wrong intersection point

You can use this function FindAllCrossings2D as follows

pts = FindAllCrossings2D[{fux[x, y], fuy[x, y]}, {x, -4, 4}, {y, -4, 
     4}, Method -> {"Newton", "StepControl" -> "LineSearch"}, 
    PlotPoints -> 256, WorkingPrecision -> 20] // Chop

Then

Show[ContourPlot[{fux[x, y] == 0, fuy[x, y] == 0}, 
  {x, -4, 4}, {y, -4,4}],ListPlot[pts]]

Note that the above link provides solutions based on MeshFunction and ListPlot3D