It also states that you cannot order the field of complex numbers. Whereas I seem to recollect that there are ways to do so. For instance, z1 < z2 if x1 < x2 or x1 = x2 and y1 < y2.
By your definition of <, 0 < i and i^2 < 0, however, the OP requires that 0 < p AND 0 < q => 0 < p * q, which is not fulfilled by your < for p = i = q.