  In this paper we describe the rise of global operators in the scaled
quaternionic case, an important extension from the quaternionic case to the
family of scaled hypercomplex numbers $\mathbb{H}_t,\, t\in\mathbb{R}^*$, of
which the $\mathbb{H}_{-1}=\mathbb{H}$ is the space of quaternions and
$\mathbb{H}_{1}$ is the space of split quaternions. We also describe the scaled
Fueter-type variables associated to these operators, developing a coherent
theory in this field. We use these types of variables to build different types
of function spaces on $\mathbb{H}_t$. Counterparts of the Hardy space and of
the Arveson space are also introduced and studied in the present setting. The
two different adjoints in the scaled hypercomplex numbers lead to two parallel
cases in each instance. Finally we introduce and study the notion of rational
function.
