Equation of the hyperbola be $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ so equation of the conjugate hyperbola is $\frac{x^2}{a^2}-\frac{y^2}{b^2}=-1$
Equation of a diameter of the hyperbola is $y=\left(b^2 / a^2 m\right) x$ which meets the hyperbola at points given by
$\frac{x^2}{a^2}-\frac{b^4 x^2}{b^2 a^4 m^2}=1 \Rightarrow x^2=\frac{a^2 m^2}{m^2-b^2}$
These points are real if $m^2 \gt b^2$ and for this value of $\mathrm{m}$, the diameter will meet the conjugate hyperbola at point for which $x^2=\frac{-a^2 m^2}{m^2-b^2}$ and these points are imaginary.
Note: Conjugate diameter will meet the conjugate hyperbola in real points and the hyperbola in imaginary points