The additive identity is easily found to be (0, 0). As for the multiplicative identity: we are looking for the complex number (u,v) that does not change anything under multiplication, i.e. for which:

(x,y)*(u,v)=(x,y)

for all (x,y). According to the definition of multiplication:

(x,y)*(u,v) = (xu - yv, xv+yu) = (x,y)

According to the definition of equality this results in a system of two equations:

(1)     xu - yv = x
(2)     xv + yu = y

Multiplying the first equation by y, the second by x gives:

xyu - y²v = xy
x²v + xyu = xy

Now we subtract the equations to get:

v(x²+y²) = 0

Since this has to be true for all real x,y we have that v=0. Substituting that in equation (1) gives u=1. Therefore, the multiplicative identity is:

(1, 0)