$$z+\bar { z } =(a+i*b)+(a-i*b)=a+a+i*b-i*b=2*a=2*Re(z)$$
$$z-\bar { z } =(a+i*bi)-(a-i*b)=a-a+i*b+i*b=2i*b=2i*Im(z)$$
$$z+\bar { z } +|z|=0$$$$\Leftrightarrow (a+i*b)+(a-i*b)+\sqrt { { a }^{ 2 }+{ b }^{ 2 } } =0$$$$\Leftrightarrow 2a+\sqrt { { a }^{ 2 }+{ b }^{ 2 } } =0$$$$\Leftrightarrow \sqrt { { a }^{ 2 }+{ b }^{ 2 } } =-2a$$$$\Leftrightarrow { a }^{ 2 }+{ b }^{ 2 }=4{ a }^{ 2 }$$$$\Leftrightarrow { b }^{ 2 }=3{ a }^{ 2 }$$$$\Leftrightarrow b=\pm a\sqrt { 3 }$$$$\Rightarrow { L=\left\{ z\in C|Im(z)=Re(z)\sqrt { 3 } \vee Im(z)=-Re(z)\sqrt { 3 } \right\} }$$