Ted's workaround was OK, but for what it is worth, here is another:
(%i40) declare(z,complex)$
(%i41) eq : [z*conjugate(z)+3*(z-conjugate(z))=13+12*%i]$
(%i42) eq : append(eq, conjugate(eq))$
(%i43) eq : subst(zc, conjugate(z), eq)$
(%i44) sol : algsys(eq, [z,zc])$
(%i45) makelist([z = subst(sk, z)], sk, sol);
(%o45) [[z=(3*%i-11)/(%i+3)],[z=2*%i+3]]
Barton