I thought of a way to use Gaussian integers to show that NxN~N We look at (1+i)(1-i) and this corresponds to the coordinate (1,1) then (1+2i)(1-2i)-->(1,2) then (1+3i)(1-3i)-->(1,3).... and you keep doing this, so we have injected NxN into N.
It looks to me like your mapping goes from N to N x N. Is that what you intended? (1 + i)(1 - i) = 1 - i^{2} = 1 + 1 = 2. So here the integer 2 is mapped to (1, 1). Did you mean for it to go the other way?
The fundamental problem is that N x N is NOT equivalent to N, it has the same cardinality as the set of rational numbers. It appears that your assignment is "one-to-one" but not "onto".