I reached solace regarding this itchy issue (not even joking) by considering the following: Primes are the minimal set of numbers to generate another set of numbers, using multiplication (with repetition, yada yada) For the positive numbers, [2, 3, 5, 7, ...] is all we need, considering that, as many others have said, 1 = Product([]).
In that sense, to generate all integers, the integers you need are: [-1, 0, 2, 3, 5, 7..]
Similarly, to generate all gaussian integers (complex numbers with integer components,) you can follow the following link: https://en.wikipedia.org/wiki/Gaussian_integer#Gaussian_prim... however, am not sure why 0 is not considered a prime number in this context.
Gaussian primes are symmetric over the real and imaginary axes, and to me it's quite curious why 0 is not considered prime there, but I guess uniqueness of the multiplication matters too (0 can be repeated multiple times, which could make some theorems about uniqueness of factorization a bit cumbersome as well.)