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My interpretation

My first question is what this type of graph (of $x-y-i$) is called since I was unable to find any information about any such graph.

Now for the real question, I used the equation $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ and drew the graph on the $xy$ plane and then added an imaginary axis and found out the value $y = bi$. so I plotted it and since it was symmetric I decided to do it for $-b$; then I saw that the values fell sharply so now I think this graph will be an Ellipse. Am I correct?

Is there any research about the xy plane with imaginary axis?

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