Suppose I have a set of values $(x, y)$ that obeys a hyperbola in the canonical form
$$ \frac{x^2}{(\sqrt{p-q})^2} - \frac{y^2}{(\sqrt{r-s})^2} = 1 \hspace{1cm}(1) $$
Where $\{p,q,r,s\}>0$ represents real parameters which can be varied at will. Suppose $p>q$ and $r>s$, we will then arrive at a regular hyperbola with intercepts $\pm \sqrt{p-q}$ on the x-axis. However, if $p<q$ and $r<s$ then we get a conjugate hyperbola with intercepts on the y-axis.
To elaborate, suppose in the latter case, $p-q=-t$ for $t>0$ and $r-s=-u$ for $u>0$. Then, my original expression for my hyperbola becomes
$$ \frac{x^2}{(\sqrt{-t})^2} - \frac{y^2}{(\sqrt{-u})^2} = 1 \hspace{1cm} (2)\\ \Rightarrow \frac{x^2}{(i\sqrt{t})^2} - \frac{y^2}{(i\sqrt{u})^2} = 1 \hspace{1cm} (3) \\ \Rightarrow -\frac{x^2}{(\sqrt{t})^2} + \frac{y^2}{(\sqrt{u})^2} = 1 \hspace{1cm} (4) $$
From equation $(4)$, this suggests that the hyperbola has been transformed into a conjugated version with y-intercept at $\pm\sqrt{u}$. My question is of the following: Based on equation $(1)$ and $(3)$, what is the correct way of interpreting the intercepts of the hyperbola? Is it correct to say that equation $(3)$ have an imaginary x-intercept at $\pm i\sqrt{t}$ for a regular hyperbola or is it the case that there exists y-intercepts of $\pm\sqrt{u}$ on a conjugate hyperbola based on equation $(4)$?