The "barrel-forearm complex" is a detailed way of describing the relationship between the barrel of the bat and the forearm. One of the areas that I have been focusing on when developing hitters is the ability to feel the barrel load with the rear forearm. For hitters who's dominant hand is their top hand, this movement mimics the same actions when working to lay back the arm during the throwing motion.
One of the most important elements in a hitters ability to control the barrel during the loading phase is is to feel the barrel tilt. This tilt, brings the barrel into a position where the wrists and forearm begin to create tension (you can see the forearm flex and lift as the barrel works upward displaying this connection.).
Looking at the video of Paul Goldschmidt below, you can see how he "sets" the barrel with a strong connection through his rear (right) forearm. There is a subtle and late "tilt" before he returns the rear elbow. This creates a controlled and violent release of the barrel around the hands where the hands act as a lever.
The more tension that is put through the wrists and into the hands, the more they can supply the "effort" needed to accelerate the barrel (load). That point, when controlled is extremely efficient.
Again, as we see in the video below with Albert Pujols, that tilt of the hands and barrel are connected and allow for an efficient return of the elbow towards the rear hip.
Equally important is the positioning of the lead forearm in a similar 90-degree position relative to the handle of the bat. What we get is a "complex" that moves fluid with the hands acting as the point to which the barrel whips around. The longer that the hitter can preserve those angles and work around the hands, the more efficient the swing. We see evidence of this in the rear elbow working down causing the lead elbow to work up matching the plane of the pitch.