We can suppose that we will create a new number system with essentially two imaginaries that do not interact. (Besides this, all quantities are taken to be integers) For example, we have an $i_1$ and an $i_2$. Then we could say
$$(a+b i_1)(c+d i_1) = ac + (ad + bc)i_1-bd$$
and, similarly for $i_2$:
$$(a+b i_2)(c+d i_2) = ac + (ad + bc)i_2-bd$$
However, for a system with $i_1$ AND $i_2$:
$$(a+b i_1+c i_2)(d + e i_1 + f i_2)=$$
$$(ad + ae i_1 + af i_2) + (bd i_1 - be + 0i_1i_2) + (cd i_2 + 0i_1i_2 -cf)$$
Above, the key thing to note is that $i_1\cdot i_2 = 0$.
I'm wondering if there is any idea or statement in math that says that I simply cannot do this. Without having worked in general systems of numbers, I'm wondering what ideas I should know about when I try to create a system like this.
Can I create this system if my main purposes are to carry out addition, subtraction, and multiplication with these numbers? Also, if I pose the additional constraint that all of these calculations are carried out modulo a prime, will this affect the system?