I just purchased a 325i and drove it 3 hours home with no issues. Stopped for gas half way started back up like normal. Parked it at home and this morning it started right up. I came in the house for 10 minutes went back out and it had stalled. Tried to start it and made one click (more like clunk).Lights come on, radio, fan...alll working. Can hear fuel pump engage. Tried jumping it thinking the extreme cold killed the brand new battery. Still the same clunk when I turn the key. Car only has valet key. The trunk button is not working. Beautiful car but currently an ornament in my yard. Not sure where to start.
On starting, you heard a clunk. If that means that the starter motor did not turn over the engine. You said the battery is new. Did the previous owner tell you that, or did you replace it yourself? If the P.O. told you that, or if the replacement battery is not a good one, have it tested ( or, just have it tested, regardless ). You can take it out and run it to a local auto parts store, they will have a tester. The starter itself could be going bad. The local parts store can test that as well, but I think they need to have the car there. One thing more, look at the cables and connectors on the battery and starter motor ( and check where they attach to the car ( ground ) ). If they are corroded, they can cause this kind of problem. The cables themselves can corrode internally. If you are handy with a multimeter, you can do some checks yourself. What is the voltage at the battery terminals with the car off? Should be 12+ volts. If you can get it to start, what is the voltage while the car is running? Should be 14+/- a bit. Carefully, put the meter on the battery terminals and have someone ( attempt to ) start the car. What happens with the voltage. You expect some drop, but if it is extreme, the starter might be attempting to pull more power than the battery can supply. Lights, radio, etc, dont take as much power as starting, so they can look OK. Another quick test, turn on the lights and try to start the car. Do the lights go very dim?
On my E34, the P.O. had routed the serpentine belt wrong, I would charge the battery, start it, run it down the road a bit, then the ABS light would come on, then others, then the car would die. The belt was slipping on the alternator pulley just a bit.
Local indie shop had the car for about a month trying to solve it. Called a small electrical specialist and he told me over the phone what to look for. Same car ( E34 ), the crank position sensor went bad, it would start, run, but die every great once in a while. Replaced it ( simple, quick disconnect and a 4 or 5 mm allen key ), and no problems until I sold her. Sniff.
On my E46 ( 2000 328i ), there is a mode you can put the instrument panel in where it will tell you many of the ECU values, including voltage. 2003 is still in the E46 family, I believe, so this may work for you. Just pay more attention to driving than voltage. https://www.bimmerforums.com/forum/showthread.php?61718-INFO-E46-Instrument-Cluster-Test
Since the PO claims the battery to be new, there's more than a good chance he didn't tighten the battery poles enough. This has happened to me on a 530i of the same generation.