Contradiction between Analytic and Numerical Integration

I think the Levin Rule is messing up somehow (the default choice of Method here). I recall seeing such a mistake somewhere before. Here's a workaround:

NIntegrate[
 f[x, p, 1*I*x]*(p^2/2), {p, -Infinity, Infinity}, {x, -Infinity, Infinity}, 
 Method -> {"CartesianRule", Method -> {"GaussKronrodRule", "Points" -> 11}}]

NIntegrate::slwcon: Numerical integration converging too slowly; suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small.

(*  0.25 + 0. I  *)

---

Addendum

A way to get the Levin Rule to work is to make the substitution p^2 -> p and use the even symmetry of f in p to change the p integral to the domain {p, 0, Infinity}. This transforms the oscillatory part of the integrand into Exp[I * <linear>], which the Levin Rule seems to handle properly.

ff[x_, p_, b_] = Exp[-p - x^2]*Exp[4 b*p/(1 + 4 b)]/Pi/Sqrt[1 + 4 b];
NIntegrate[
 ff[x, p, 1*I*x]*(Sqrt[p]/2),
 {p, 0, Infinity}, {x, -Infinity, Infinity}]
(*  0.25 - 5.07359*10^-13 I  *)

If we nest the NIntegrate it gets it right..

 c=2;
 g[x_?NumericQ] := 
        NIntegrate[f[x, p, c*I*x]*(p^2/2), {p, -Infinity, Infinity}]
 NIntegrate[g[x], {x, -Infinity, Infinity}]

0.25 + 0. I

however.. reversing the order (which shouldn't matter) fails:

h[p_?NumericQ] := 
 NIntegrate[f[x, p, c*I*x]*(p^2/2), {x, -Infinity, Infinity}]
 NIntegrate[h[p], {p, -Infinity, Infinity}]

(convergence warning)

0.665207 - 0.0424908 I