I don't see any reason why it isn't possible.
It is possible--it's just simply less frequently encountered. There have been a number of posts (e.g., induction on real numbers) that deal with induction as not related to natural numbers. There have also been quite a few articles, some open source and some professionally published (not mutually exclusive), published about the matter. I recall one appearing the College Mathematics Journal not all that long ago.
Concerning your own question about something like $P(-1)$ or integers in general, there was a question a while back asked about using a negative number as a base case. The short answer is yes, you can use a negative number in your base case(s). It's just slightly atypical. For anyone not interested in seeing the answer on the other thread, below is an example of proving a claim for all $n\geq -5$ by induction.
Example: Suppose you have the statement $S(n)$ where
$$
S(n) : n+5\geq 0,
$$
and you claim this is true for all $n\geq -5$, where $n\in\mathbb{Z}$. Your base case would be $n=-5$, and this is true since $-5+5=0\geq 0$. As explained above, when we reformulate the proposition, the base case would be $T(1) = 0 = S(-5)=S(k)$. The following schematic may be easier to understand:
$$
\color{blue}{T(n)}\equiv S(n+k-1) : (n+k-1)+5\geq 0\equiv n+k+4\geq 0\equiv\underbrace{\color{blue}{n-1}}_{k\,=\,-5}\color{blue}{\geq 0}\\\Downarrow\\[1em] \color{blue}{T(n): n-1\geq 0}.
$$
As you can see above, proving $S(n)$ is true for all $n\geq -5$ is the exact same as proving $T(n)$ is true for all $n\geq 1$.