  In this paper, we first provide an updated survey of the geometry of complex
Cartan spaces. New characterizations for some particular classes of complex
Cartan spaces are pointed out, e.g. Landsberg-Cartan, strongly Berwald-Cartan
and others. We introduce the Cartan-Randers spaces which offer examples of
Berwald-Cartan and strongly Berwald-Cartan spaces. Then, we investigate the
complex geodesic curves of a complex Cartan space, using the image by Legendre
transformation ($\mathcal{L}-$ duality) of complex geodesic curves of a complex
Finsler space. Assuming the weakly K\"{a}hler condition for a complex Cartan
space, we establish that its complex geodesic curves derive from
Hamilton-Jacobi equations. Also, by $\mathcal{L}-$ duality, we introduce the
corespondent notion of the projectively related complex Finsler metrics, on the
complex Cartan spaces. Various descriptions of the projectively related complex
Cartan metrics are given. As applications, the projectiveness of a complex
Cartan-Randers metric and the locally projectively flat complex Cartan metrics
are analyzed.
