What would be the implications of assuming the existence of an imaginary number that can divide a prime number and is related to the number it is dividing? By imaginary I mean a number that is just in our imagination and not the imaginary number "i".
The question is a little weird. Does it have to be an imaginary number or can it be any complex number? For example 2 = (1+i)(1-i). The Gaussian integers which are all numbers of the form a+bi for a and by integers form a ring which can be used to deduce some interesting number theoretic facts, for example every prime equivalent to 1 mod 4 can be expressed as the sum of two squares http://en.wikipedia.org/wiki/Gaussian_integer
I don't mean the imaginary number "i". I meant to say some thing that is just an idea - an imagination.