For this I'd probably use just the old L297 and L298 combination. Eight coils would be like a bipolar stepper field and the rod magnet would be like the rotor of the motor. Although there are a few three pole stepper drivers out there they aren't too easy to find.Why 1 more set of coils? Can you elaborate?
The stepper motor driver idea crossed my mind too. Do you have a particular chip in mind? The advantage would be that the driver bridge is already built-in. No external MOSFETs necessary.
However, are you referring to the chip you recently purchased for your project? That has 2 fullbridges, right?
He's demonstrating a stepper motor used in a car so this would be perfect.