First, start by doing a few quick substitutions:

x = a + bi (where a and b are real)

x* = a - bi

z = c + di (where c and d are real)

z* = c - di

xx* = (a + bi)(a - bi) = a^2 + b^2

z - z* = c + di - (c - di) = 2di

So:

a^2 + b^2 + 3(2di) = 13 + 12i

Since i is the only possible imaginary value (all other variables must be real):

a^2 + b^2 = 13 and 6di = 12i

di = 2i, and since d must be real, d = 2

So z = c + 2i, where c is any real number. The simpilest solution is z = 2i (c = 0)