In this paper, we study certain sectional structures of the t-scaled hypercomplex numbers \(\mathbb{H}_{t}\) for a scale \(t\in \mathbb{R}\) , including the quaternions \(\mathbb{H}_{-1}\) , and the split quaternions \(\mathbb{H}_{1}\) . For a fixed scale \(t\in \mathbb{R}\) , by defining the collection \(\mathbb{S}_{t}\) of certain pure-imaginary t-scaled hypercomplex number in \(\mathbb{H}_{t}\) , we sectionize \(\mathbb{H}_{t}\) from the imaginaries of \(\mathbb{S}_{t}\) . We concentrate on a section \(\mathbb{SH}_{I_{t}}\) for an arbitrarily fixed imaginary \(I_{t}\in \mathbb{S}_{t}\) , called the t-scaled section for \(I_{t}\) . Differentiation theory on the section \(\mathbb{SH}_{I_{t}}\) is studied in terms of that on \(\mathbb{H}_{t}\) by regarding \(\mathbb{SH}_{I_{t}}\) as a sub-structure of \(\mathbb{H}_{t}\) . Also, some functional vector spaces induced by \({\mathbb{SH}}_{I_{t}}\) over the real field \(\mathbb{R}\) are constructed and analyzed. And then an interesting type of operators on one of our vector spaces is considered. In particular, we are interested in Toeplitz-like operators. The main tool to do them is the isomorphic relation between \({\mathbb{SH}}_{I_{t}}\) and the t-scaled hyperbolics \(\mathbb{D}_{t}\) , for “all” \(I_{t}\in \mathbb{S}_{t}\) , for any scale \(t\in \mathbb{R}\) .
