First of all, I would not call it the "imaginary axis" - the name needs to be clearer as to what the value on that axis represents. If I understand your question correctly, it is the imaginary $y$-axis; in other words, you're allowing $y$ to be a complex number $y=y_0+iy_1$, while you are still requiring $x$ to be real.
If I understand your question correctly, you are wondering if your plot of the solutions to $\frac{x^2}{a^2}-\frac{(y_0+iy_1)^2}{b^2}=1$ is correct. Well, I'd advise thinking about it like this: $\frac{x^2}{a^2}-\frac{(y_0+iy_1)^2}{b^2}=\bigg(\frac{x^2}{a^2}-\frac{y_0^2}{b^2}+\frac{y_1^2}{b^2}\bigg)-i\bigg(\frac{2y_0y_1}{b^2}\bigg)=1$ The only way this is possible is if $y_0=0$ or $y_1=0$; otherwise, the imaginary part of the left side is non-zero, while the imaginary part of the right size is zero.
Thus, you're looking for the solutions to $\frac{x^2}{a^2}-\frac{y_0^2}{b^2}+\frac{y_1^2}{b^2}=1$ where either $y_0=0$ or $y_1=0$. The above equation defines a hyperboloid of one sheet, and so you're looking for the intersection of that hyperboloid with the $xy_1$-plane (where $y_0=0$) and $xy_0$-plane (where $y_1=0$).


In conclusion: Your plot seems to be of the right form, though it is not centered correctly (the center of the whole thing should be at the origin).