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The joint characteristic function of the real and imaginary parts of $Z$ is derived in: Distribution of Inner Product of Complex Gaussian Random Vectors and its Applications (2011). That publication is behind a pay wall. You can find the expression in equation 14 of this paper (set all the $q_i$'s equal to unity).

The joint characteristic function $\Psi(\omega_1,\omega_2)$ of the real and imaginary parts of $Z=z_1+iz_2$ is derived in: Distribution of Inner Product of Complex Gaussian Random Vectors and its Applications (2011). That publication is behind a pay wall. You can find the expression in equation 14 of this paper, for the most general case that each $x_i$ and $y_i$ can have different Gaussian distributions. For the case in the OP the characteristic function is simply $\Psi(\omega_1,\omega_2)=[1+\tfrac{1}{4}c^2(\omega_1^2+\omega_2^2)]^{-M}$.

The joint distribution of the real and imaginary parts of $Z$ is derived in: Distribution of Inner Product of Complex Gaussian Random Vectors and its Applications (2011). That paper is behind a pay wall. You can find the expression in equation 14 of this open access publication.

The joint characteristic function of the real and imaginary parts of $Z$ is derived in: Distribution of Inner Product of Complex Gaussian Random Vectors and its Applications (2011). That publication is behind a pay wall. You can find the expression in equation 14 of this paper (set all the $q_i$'s equal to unity).

The joint distribution of the real and imaginary parts of $Z$ is derived in: Distribution of Inner Product of Complex Gaussian Random Vectors and its Applications (2011).

The joint distribution of the real and imaginary parts of $Z$ is derived in: Distribution of Inner Product of Complex Gaussian Random Vectors and its Applications (2011). That paper is behind a pay wall. You can find the expression in equation 14 of this open access publication.