I think the thing that you are missing is the distinction is not at all clear cut and that there are many instances where the two subjects are interwoven and blend more or less seamlessly, both historically (starting from the amazing fact that compact Riemann surfaces are algebraic and the Lefschetz principle showing that complex algebraic geometry gives insight into algebraic geometry over any field of characteristic 0) and in contemporary research.

A (stereo)typical complex algebraic geometer, with no number-theory/arithmetic geometry sideline, is interested in understanding complex projective algebraic varieties, their geometry/topology, their birational classification, their algebraic cycles, their moduli, their vector/principal bundles, their enumerative geometry, their link with mathematical physics, symplectic geometry, representation theory, etc. For this, (s)he is willing to use all the tools at their disposal:

\- topology: singular cohomology and fundamental groups mostly, characteristic classes often, but also sometimes topological K-theory, complex cobordism...

\- analysis: basic differential geometry, hermitian metrics, L\^2-cohomology, Hodge theory, potential theory...

\- algebraic analysis: D-modules with regular and irregular singularities, the Riemann-Hilbert correspondence, microlocal sheaf theory...

\- reduction to positive characteristic in general, and finite fields in particular: point counting and the Weil conjectures to compute Betti numbers over C, Mori's proof via Bend-and-Break that Fano varieties are rationally connected...

\- pure Grothendieck style AG: defining moduli functors and studying their representability by schemes/stacks, , even using some derived algebraic geometry to correctly define enumerative counts...

and much much more besides.

Of course any one researcher or any one paper on the subject is not going to showcase all the techniques but there is a profound unity to the subject.