For complex numbers it's fairly intuitive. i² corresponds to a nice counter-clockwise rotation along the unit circle.
A "rotation" in the split-complex plane however corresponds to a movement alongside the "unit-hyperbola(s)". This is equivalent to a squeeze-mapping.
As you can see there's no hyperbola on which $j$ rotates elements that leads from j to +1. What does this mean? How can we understand why j² is 1?
This is more of a comment than an answer, but it's too long. Multiplication by $j$ corresponds to reflecting points across the diagonal (the $y = x$ line) of the plane; $j \cdot (x + j y) = y + j x$.
Here's a relationship between multiplication and rotation. A null-basis for the split-complex numbers is given by $u = \frac{1 - j}{2}$ and $\bar{u} = \frac{1 + j}{2}$. For $z = z^u u + z^{\bar{u}} \bar{u}$ and $w = w^u u + w^{\bar{u}} \bar{u}$, $z\cdot w = z^u w^u u + z^{\bar{u}} w^{\bar{u}} \bar{u}$; i.e., the split complex numbers are isomorphic to the ring $\mathbb{R} \times \mathbb{R}$. Since $\cosh \theta + j \sinh \theta = e^{-\theta} u + e^{\theta} \bar{u}$, one finds $(\cosh \theta + j \sinh \theta) \cdot (\cosh \phi + j \sinh \phi) = \cosh(\theta + \phi) + j \sinh(\theta + \phi)$.