# Notion of line on a complex affine space


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?