Battlemage grants Added Spell Damage equal to the Damage of your Main Hand Weapon. This provides a lot of new hybrid spellcaster-melee build possibilities.
Sources: Instruments of Virtue(Notable ascendancy passive skill from Inquisitor class), Martyr of Innocence(unique staff, requries level 52), and Disintegrator(unique staff, requires level 64). Note that the effect of Battlemage does not stack.
1. Battlemage Build
While building Battlemage skill, always remember its tag: Spell Damage.
Let's choose Disintegrator as the main hand weapon. It grants Battlemage and also has high physical damage.
Main hand weapon: Disintegrator
Main skill: Vaal Blade Vortex
Class: Pathfinder
Vaal Blade Vortex tag: Vaal, Spell, AoE, Duration, Physical. Its spell and physical damage tags are the best tags for Battlemage.