The quadratic equation$p \left(x\right) = 0$ with real coefficients has purely imaginary roots. Then the equation $\rho \left[p\right(x\left)\right]=0$ has

If quadratic equation has purely imaginary roots, then coefficient of x must be equal to zero. 

Let $p\left(x\right)=a{x}^2+b$ with a, b of same sign and $a,b\in R$

Then, $p\left[p\right(x\left)\right]=a{\left(a{x}^2+b\right)}^2+b$

$p\left(x\right)$ has imaginary roots say ix. 

Then, also $a{x}^2+b\in R\text{ and }{\left(a{x}^2+b\right)}^2>0$

$\therefore a{\left(a{x}^2+b\right)}^2+b\ne 0,\mathrm{\forall }x$

Thus,  $\rho \left[\rho \right(x\left)\right]\ne 0,\mathrm{\forall }x$