Are there Olympiad problems that can be solved using split-complex numbers?

Question

There are various uses for complex numbers in Math Olympiads. In addition, quadratic number fields are sometimes useful, for instance for Pell equations.

Are there any Olympiad/contest problems that can be solved using split-complex numbers ($a+jb$, $j^2=1$)?

Answer 1

Just made this up.

Show that the product of 2 integers, each of which can be written as the difference of 2 squares, can again be written as the difference of 2 squares.

The algebraic approach would involve showing

$$ ( a + bj) ( c + dj) = (ac+ bd) + (ad + bc) j. $$

Of course, the number theoretic approach is slightly more direct. We use the classification that a number $n$ can be written as the difference of 2 squares iff it is either 1) odd, or 2) a multiple of 4.