Assuming you are not "steering" and the putter head is traveling along its undisturbed orbit, then the putter head will be traveling forward, out and down until low point (opposite the left shoulder in Homers putting method elsewhere for others). Past low point it will be traveling forward, in and up.
I think there are a lot of golfers who understand the beauty of 3D impact for their full shots but steer their putter along a straight line. Watching the putter head going back and through can be dangerous in this regard.
I heard that Homer putted a very worn carpet remnant in his garage. Dont know if this influenced his putting style or not. Im sure he must have putted the shaggy greens of the 60's and 70's too.