To start you off this works:
InterpolatingPolynomial[{{{4, 4}, 52}, {{4, 6}, 120}, {{4, 8},
212}, {{6, 4}, 120}, {{6, 6}, 256}, {{6, 8}, 432}, {{8, 4},
212}, {{8, 6}, 432}, {{8, 8}, 708}, {{10, 6}, 648}, {{10, 8},
1040}}, {w, z}]//Expand
(4 - w^2 - 3 w z + w^2 z - z^2 + w z^2)
Maybe you did not have enough points before to uniquely determine all the coefficients.
The above answer is not the best way to handle this. When you know you are right and Mma is not then you should roll your own:
fun[l_] := Module[{w, z},
w = First[l[[1]]];
z = Last[l[[1]]];
h + g*w^2 + f*w^3 + e*w* z + d*w^2* z + c*z^2 + b*w* z^2 + a*z^3 ==
l[[2]]]
pts = {{{4, 4}, 52}, {{4, 6}, 120}, {{4, 8}, 212}, {{6, 4},
120}, {{6, 6}, 256}, {{8, 4}, 212}, {{8, 6}, 432}, {{8, 8}, 708}};
rul = Solve[fun[#] & /@ pts, {a, b, c, d, e, f, g, h}]
(* {{a -> 0, b -> 1, c -> -1, d -> 1, e -> -3, f -> 0, g -> -1, h -> 4}} *)
(h + g*w^2 + f*w^3 + e*w* z + d*w^2* z + c*z^2 + b*w* z^2 +
a*z^3) /. rul
{4 - w^2 - 3 w z + w^2 z - z^2 + w z^2}
Which is what you want. This does not require any changes to the data so disregard my earlier solution.