A common convention in many algebras is that modulus is
$||z||=\exp( R (\ln z))$
where R(z) is the real or scalar part.
If we use this convention for split-complex numbers, we have:
$$\ln (a+bj)=\frac12 \ln ((a+b)(a-b))+j\frac12 \ln\left(\frac{a+b}{a-b}\right)$$
The real part is $\frac12 \ln (a^2-b^2)$, so the modulus should be $\sqrt{a^2-b^2}$.
Yet, according to Winkipedia it is $a^2-b^2$. Why?