The paper Kötter's synthetic geometry of algebraic curves, (N. Fraser, Proceedings of the Edinburgh Mathematical Society 7, 46–61, 1888) opens with a sketch of what appears to be a synthetic construction of the complex projective plane from the real one. "Imaginary points" are defined to be projective involutions on real lines without real fixed points. This looks very cool, but there are a lot of details left out; is it written down carefully anywhere?
Edit: Here is a quote to give more of the idea:
An elliptic involution has, we know, no real double elements. Imaginary points in a line are defined as the double points of the various elliptic involutions which may be formed of the real points of the line. Each involution thus yields a pair, an imaginary point and its conjugate, which, like the involution itself, are completely determined when two real pairs of the involution, say $AA'$ and $BB'$, are given. These pairs will divide one another, $B$ and $B'$ lying the one in the finite, and the other in the infinite line $AA'$. We denote the two imaginary points which they determine by writing them down in order, as $ABA'B'$. It remains to distinguish between the two imaginary points thus denoted... by the order in which we name the four points which determine the involution; and if we denote the one by $ABA'B'$, we denote the other by $AB'A'B$...
In exactly the same way imaginary lines, through a real point, are defined as the double rays of an elliptic involution in a pencil, and are denoted by $aba'b'$, $ab'a'b$, respectively.
An imaginary line is said to contain an imaginary point, if it be possible to represent the one by $aba'b'$ and the other by $ABA'B'$, where $a$, $b$, $a'$ and $b'$ are the lines joining $A$, $B$, $A'$ and $B'$ to $O$, the real point of the imaginary line.