Can an ellipse be thought of simply as a hyperbola in the complex plane? I have been doing some problems from textbooks in ellipses and hyperbolas and very often I have noticed a slightly uncommon trick to remember tangent, normal formulas of ellipses and hyperbolas. While writing the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ simply replace the $b$ here in the ellipse by $\iota b$. That does the trick in most of the formulas. However when I tried various programs for plotting this curve $\frac{x^2}{a^2}+\frac{y^2}{(\iota b)^2}=1$, it did not let me. I was wondering if replacing this by iota takes the curve away from the real plane all the way to the imaginary plane making it look like a hyperbola in it. I want an answer to whether what I am thinking is indeed true, or is it just a trick to remember formulas and it does not relate to something beautiful in any way.
Any recommendation of programs where I could plot such type of a curve would also be appreciated.
 A: 
Can an ellipse be thought of simply as a hyperbola in the complex plane?

I'd say that depends.
If by complex plane you mean $\mathbb C^1$, the plane of complex numbers, then no, that doesn't really fit. So I'll ignore this interpretation for the rest of my answer.
If by complex plane you mean $\mathbb C^2$, a complex two-dimensional space, then I'd say that in that world, you have to note that the resulting object has four real dimensions. That makes it really hard to imagine, thus defying intuition.
On a purely algebraic level you could say that in this complex plane the distinction between ellipse and hyperbola becomes moot. Typically you'd distinguish these shapes by looking at the sign of some discriminant or other, but when the discriminant is complex-valued then the simple concept of a sign becomes the more complicated concepts of a complex argument, an angle. So both ellipse and hyperbola would be elements of a more general class of things, and there would be a continuous rotation of these general objects which have ellipse and hyperbola as special cases, special positions. About as special as an axis-aligned ellipse would be in the real plane, if you want a comparison.
Or you could take slices. Intersect your $\mathbb C^2$ with a real plane and look at the section that plane forms with an otherwise complex curve. If you define your plane as spanned by the real $x$ axis and the real $y$ axis, you get the conventional plane, but if you intersect with the real $x$ axis and the imaginary $y$ axis, or vice versa, then you get a world where the roles of ellipses and hyperbolas can be swapped.
You can also do a 3d cut. Take the real $x$ axis, the real $y$ axis and the imaginary $y$ axis as three spatial dimensions. Now you can visualize things to some degree. For example, if you take the unit circle (as a very special ellipse) you will find that you get two possible $y$ values for most $x$ values. For $-1<x<1$ those would be purely real, for $\lvert x\rvert>1$ purely imaginary. You can see aspects of the circle in such a drawing that are hard to grasp in 2d. But you need to still remember that you are looking at a somewhat arbitrary slice of a larger four-dimensional space, and are missing anything with a non-zero imaginary part for $x$.
Projecting 4d space into a 2d should be possible in theory as well, but I'd be worried you'd loose too much information for this to be of any use. Haven't tried, though. Since a quadric would be a 2d surface in 4d space, I expect a projection would be a boring region of the plane unless you add some form of lighting to give a more spatial impression. Tricky, but projecting from 4d to 3d first might allow you to delegate that to some existing tool.
