The center of the circle represented by $\left|z+1\right|=2$ $\left|z-1\right|$  on the complex plane is:

    $\left|z+1\right|=2\left|z-1\right|$ 

$\Rightarrow$       ${\left|z+1\right|}^2=4{\left|z-1\right|}^2$

$\Rightarrow$     $\left(z+1\right)\left(\overline{z}+1\right)=4\left(z-1\right)\left(\overline{z}-1\right)$

$\Rightarrow$        $z\overline{z}+z+\overline{z}+1=4z\overline{z}-4z-4\overline{z}+4$

$\Rightarrow$        $3z\overline{z}-5z-5\overline{z}+3=0$

$\Rightarrow$       $z\overline{z}-\frac{5}{3}z-\frac{5}{3}\overline{z}+1=0$

$\therefore$                         Center $=\frac{5}{3}$