Let $d>0$ be an integer and consider the first neighbors independent bond percolation model in $\mathbb Z^d$, where each edge is open with probability $p\in[0,1]$. I would like to know, if can we generalize the construction of the probability measure of this model for a complex parameter $p\in \mathbb C$ in some neighborhood of zero.

In other words,

Let be $E=\{\{x,y\}\subset\mathbb Z^d: \|x-y\|_1=1\}$, where $\|x-y\|_1=\sum_{j=1}^d|x_j-y_j|$.

If $\{0,1\}$ a complex measure space, such that $\mu(\{1\})=p\in\mathbb C$ and $\mu(\{0\})=1-p$, is there any domain $D$ on the complex plane, for which it is possible to construct a product measure, formally given by

$$\mathbb P_p=\prod_{e\in E} \mu $$

defined on the sigma algebra generated by the cylinder sets of $\Omega=\{0,1\}^{E}$ with $\mathbb P_p(\{0,1\}^{E})=1$

for any $p\in D$ ?

If the answer is no. What is the best set-function fitting as much as possible the properties of a measure we could construct for a $p$ in some domain of the complex plane ?

This question it was motivated by the possibility, in case of a positive answer ( or a good construction near a probability), to use complex analysis results, to study the behavior of the probability of certain events as a function of $p$.