To start from the beginning my original alternator seized on my 96 coupe and that alone was a bastard to get out of the car put the new alternator in the car. At this point when I turned the key the battery light would illuminate then when I’d start the car it was still on but dim. So I went and got the alternator tested and it passed. I determined the belt was slipping causing the dim light however I cannot verify because after the second installation the battery light no longer illuminates prior to start or at all. I thought maybe the bulb had burned out however I switched the oil pressure bulb with the battery bulb and the bulb is not burned out... I guess I’m curious what it could be car runs and drives fine

 

My alternator went out as well. I installed a new one and a battery. My damn battery light is on and like yours very dim. Its so damn annoying and I tested everything. I have done deep research on this and can't find barely anything. I have a Denso alternator because they are supposed to be good for Honda's. I tested the fuses, all the voltage and amperage ground wires all 3 of my drive belts everything and it all checks out. I may have to get another alternator. I dont even know what a dim battery means. I have asked a bunch of people and no one has a solid answer. A fully light Battery light is bad that much I do know, but a dim battery light I have no clue. My car runs fine with no issues. That battery light is driving me insane.

 

Check whether 7.5A dash fuse 15 is blown.

 

 

With key in ON(II) [engine off], check whether whether voltage fails to reach the socket for charging system bulb in the cluster.

You can also check whether the bulb turns ON if you unplug the 4P alternator connector and ground the Wht/Blu wire.