Hypercomplex Numbers of the Form $a + ib$ Where $i^2 = p + iq$

Question

Recently, I've been interested in hypercomplex number systems, and I have come across the three main 2-dimensional algebras: \begin{align*} i^2 &= -1 \\ j^2 &= 1 \\ \varepsilon^2 & = 0 \end{align*} I know that there are many more well-studied systems in higher dimensions, but I am more interested in basic two-dimensional algebras. These algebras would contain a single numbers with a single real part and imaginary part (multiplied by $i$) where $$ i^2 = a + ib $$ My question is why aren't the other possible number systems of this form as popular as the complex, split-complex, and dual numbers? $i^2 = k$ seems to hold similar properties to the other systems, and $i^2 = 1 + ik$ holds similar properties to the metallic means. Are the other systems not as useful as the complex numbers, etc., or do they just not hold basic properties that one would want an algebra to have?