Best charge?

I decided to help a co-worker out. Her car was diagnosed by her neighbor he said it need an altenator. I go to her house and I check everything, I see that the battery has less than 6 volts I jump the car and I get a charge of around 12.5 volts. I told her that I need to replace the battery, and maybe the altenator down the road. I install the new battery and start the car and now I am getting 13.4 volts of charge this is perfect isnt it? But the battery light is on and the car is running rough? No check enging light or anything else just the battery light help if you can!

Year, make and model of car?
If it’s a fairly new car, it’s usually the computer resetting itself. Also, check the cables for the battery, they might need cleaned.

Battery light still on may mean a bad alternator.

Get a full load test done on the complete charging system.

A lot could depend on the type of car, mileage, etc. Internal or external voltage regulator, etc.

13.4 volts is not perfect under all conditions. If there are accessory items on (any lighting, heater/ac fan, etc) or a voltage drain somewhere then 13.4 could be about right.
With a new battery and everything off except what is essential to running the engine, you should have a voltage reading of around 14.

I agree 14.3 volts is questionable.

If -by tiny chance- this is a 97 Explorer check mini fuse 6 under the hood in the power distribution box.

Strangely enough Ford labels this fuse “Generator/voltage regulator” & if it’s blown it wont charge.

Take a look at the fuse list in the owners manual.

OOPS, typo 13.4 volts is questionable

93 toyota corolla 1.6L