A Wick-rotation in spacetime $x^{\mu}$ implies via Fourier transformation a Wick rotation in Energy-momentum space $p_{\mu}$. Perhaps the easiest way to convince oneself that this must be so is to consider the Fourier-integral representation $$\delta^4(x)~=~\int_{\mathbb{R}^4} \frac{d^4p}{(2\pi\hbar)^4}~\exp\left(\frac{ip\cdot x}{\hbar} \right)\tag{A}$$ of the Dirac delta distribution. It cannot be analytically continued to the ambient complexified spacetime. The real integration region can at most be deformed, i.e. the $x^0$ and $p_0$ Wick-rotations must be balanced.

A Wick-rotation in spacetime $x^{\mu}$ implies via Fourier transformation a Wick rotation in Energy-momentum space $p_{\mu}$. Perhaps the easiest way to convince oneself that this must be so is to consider the Fourier-integral representation $$\delta^4(x)~=~\int_{\mathbb{R}^4} \frac{d^4p}{(2\pi\hbar)^4}~\exp\left(\frac{ip\cdot x}{\hbar} \right)\tag{A}$$ of the Dirac delta distribution. It cannot be analytically continued to the ambient complexified spacetime. The real integration region can at most be deformed, i.e. the $x^0$ and $p_0$ Wick-rotations must be balanced. See also e.g. this and this related Phys.SE posts.