The correct option is C a pair of straight lines
z2+z|z|+|z|2=0⇒(z|z|)2+z|z|+1=0
Assuming z|z|=t
⇒t2+t+1=0⇒t=ω,ω2⇒z|z|=ω,ω2⇒z=ω|z| or z=ω2|z|⇒x+iy=|z|(−12±i√32)⇒x=−|z|2, y=±√3|z|2
Squaring both the sides,
4x2=x2+y2, 4y2=3(x2+y2)⇒3x2−y2=0
This represents a pair of straight lines.