first of all, on , |z| does not mean the absolute value of z, but rather the modulus (distance from the origin) of the complex number z.
so any of your arguments that use "absolute value" are null and void.
it IS true, however, that |zw| = |z||w| (you should PROVE this!), and this fact can be used to show associativity.
i don't think you understand what a group identity IS. it is an element e for which x*e = e*x = x for ALL x.
consider the complex number 1 = 1+0i. what is 1*z ( = |1z|)...? is this equal to z if z = -1? how about if z = √2/2 + i(√2/2)?
even as it stands, your argument on inverses is flawed. inverses must be UNIQUE.
for example, consider the complex number 4 = 4+0i. what is 4*(1/4), and 4*(-1/4)? is 1/4 a unique inverse for 4?