Fourier-like transformation with split-complex numbers?

Question

Usually the Fourier transformation is defined using the imaginary unit $$ F(p) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} f(x) e^{ipx} dx = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} f(x) \cos px + i \sin px\ dx $$ Now with split complex numbers one could come up with the idea to replace the imaginary unit $i^2=-1$ with $j^2=1$ and get in analogy a kind of hyperbolic Fourier transformation $$ F_h(p) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} f(x) e^{jpx} dx = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} f(x) \cosh px + j \sinh px\ dx $$ Does such a transformation make sense? Would it have any use?