Disclaimer
I know that in a "classical" sense the infinite product of negative ones is not defined. However, there are many other "senses" we are aware of, so maybe it is the case. This is how this idea was generated in my mind: considering the infinite sum of consecutive powers of negative ones --- what is its analogy in the world of infinite products?
Let's consider the infinite product of negative ones (whatever the $:=$ sign means): $$ \mathcal{P} := \prod_{n=1}^{\infty}(-1) $$ I found that "in some sense" it is equal to negative imaginary one $$ \mathcal{P} = -\mathrm{i}\label{eq}\tag{1} $$ How? Let's consider two sequences $$ \mathcal{P}_N = \prod_{n=1}^{N}(-1)\\ \mathcal{Q}_N = (-1)^{\sum\limits_{n=1}^N(-1)^n} $$ it is easy to see that for all $N>0$: $\mathcal{P}_N = \mathcal{Q}_N$.
"Proof" of \eqref{eq}
The following chain of equalities (probably all of them but last are incorrect) proves the result: $$ \mathcal{P} = \lim_{N\to\infty}\mathcal{P}_N = \lim_{N\to\infty}\mathcal{Q}_N = (-1)^{\lim\limits_{N\to\infty}\sum\limits_{n=1}^{N}(-1)^{n}} = (-1)^{-\frac{1}{2}} = -\mathrm{i}. $$ Here I used the result that $$ 1 - 1 + 1 - 1 + \dots = \frac{1}{2} $$ which is also seems like a fake result but read this.
Question
Is there some sense behind this result? I mean is this result make sense somewhere in advanced mathematics? Maybe there is an analytic function that makes this result valid due to its analytical continuation or smth. For example, like the result that $1+2+3+4 + \dots=-1/12$ for zeta-function.
We can extend your idea to a general $k$-th root of unity. Take this root to be $e^{i\theta}$. Then we can write
$$\mathcal{P}_N = \left(e^{i\theta}\right)^{\sum_{n=1}^{N} a_n}$$
where the sequence of partial sums $\sum_{n=1}^N a_n = N \operatorname{mod} k$. While the summand oscillates, hence the summation does not converge, we can assign it a Cesaro sum (i.e. an average) of $\frac{k-1}{2}$.
Thus the limit can be assigned the value of
$$ \mathcal{P}_\infty = e^{\frac{i (k-1) \theta}{2}}$$
But depending on how the sum is rearranged, it can have any value $ \frac{n}{2}$ for integer $|n| < k$. This corresponds to how each root of unity technically evaluates to different limits even though they are multiplying the same terms.