4 years ago, when I learned about factorising and complex numbers, me and my friend worked on factorising complex numbers.
For example, $ 4+2i= 3-(-1)+2i = 3-i^2+2i = -(i^2-2i-3) = -(i-3)(i+1) $
The goal was to represent $a+bi$ with product of same form. where $a$ and $b$ are integer.
Another example is, $ 8+i= 8i^4+i=i(8i^3+1)=i(2i+1)(4i^2-2i+1)=i(2i+1)(-2i-3)=-i(2i+1)(2i+3) $
I showed my teacher, and she said it's useless.
Now I think of it, I don't know why I did this and it looks like same thing just in different form.
Is there any research already done on this or can there be any use of it?