$\newcommand{C}{\mathbb{C}} \newcommand{\A}{\mathbb{A}} \newcommand{\i}{\hspace{0.1em}\mathrm{i}} \newcommand{\R}{\mathbb{R}}$ Let $V = (\C, \C, +, \cdot)$ be the one-dimensional complex vector space over the complex field, where $+$ and $\cdot$ are the usual complex addition and multiplication, respectively. Let $\A_\C$ be the affine space $(\C, V, \delta)$, where $\delta$ is the (canonical) map $$ \begin{align} \delta\colon\quad \A_\C \times \A_\C &\to V\\ (A, B) &\mapsto \overrightarrow{AB} = B - A\,. \end{align} $$

Lines are defined, in affine geometry, as one-dimensional affine subspaces. Since $V$ is formally one-dimensional, is it true then that $\A_\C$, the whole "complex plane", is technically a line?

If so, what would be the extension of the notion of a segment to such affine space? In real affine spaces, the segment between two points $A, B$ is defined as the set of points $$ \overline{AB} = \{A + \lambda\overrightarrow{AB} \mid \lambda\in[0, 1]\}\,. $$ In the aforementioned complex affine space, would the set $$ \{A + (a + b\i)\overrightarrow{AB} \mid a, b \in [0, 1]\subseteq\R\} $$ be a natural extension of this concept? What would this set look like?


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