# Notion of line on a complex affine space

$$\newcommand{C}{\mathbb{C}} \newcommand{\A}{\mathbb{A}} \newcommand{\i}{\hspace{0.1em}\mathrm{i}} \newcommand{\R}{\mathbb{R}}$$ 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?