Yeah it sounds like a bad coil to me, try cleaning the coil and check the shut off wire to make sure its properly grounded. Then make sure the gap between the coil and flywheel is .010-.014 inches. Replace the plug and gap that at .30 inches and let your mower fun for about an hour. If the mower dies then replace your coil for about $60-$80.

When it dies, do you have any spark to the plug? And, is there any fuel reaching the carburetor? What condition is the fuel line in? See if it might have a soft spot where the line might suck closed. Check the breather hole in the fuel cap. I had an old Wheel Horse that the line would suck closed, then reopen after sitting for ten minutes. Also, check for dirt in the fuel bowl, or water in it. In a few cases, the ignition coil/magneto fails from heat when running, but works again after it cools off. Thats why i say to check the spark immediately after it stops!

Huh, kinda lucky it ran at all, I'd say. Must've been weak fuel pressure, but still functioning somewhat? I'm curious, but did you ever remove the original fuel pump from the tank and check it out? Your 445 has an internal pump (inside the fuel tank) doesn't it? I wonder if the filter sock had plugged up or come off and allowed the pump to suck up junk from the bottom of the fuel tank? Somehow, people always seem to get "crap" in their fuel tanks, leading to future problems. Glad to hear you came up with a cheaper fix with the automotive style pump. It may fail sooner, but it's hard to say. External electric fuel pumps typically fail sooner than internal ones due to heat. What most people fail to realize is that fuel pumps run/stay cooler when immersed in fuel. (That's really the only reason why they put them inside the fuel tanks in the first place.) My JD 322 has a external in-line fuel pump. The original JD pump failed after 12 years of service, and it acted much like your 425 did. I too, went to the auto parts store and got an inline pump that was nearly identical to my original one, for about 1/3rd the cost of what my JD dealer wanted for the same pump. How aggravating that was, that JD's price was so unreasonable! Anyhow, the only thing I had to change was the plug connector, in which I cut the original one from the old pump and connected it to the new one so that it would fit exactly as the old pump did. It has now been on my tractor and working great for the past 6 years with no problems. Yee-haw! Oh, the only other thing I was going to mention was this... These days, many small engines have electric fuel shut-off solenoids on the carburators. When they fail, they either fail partially - leading to power loss issues, or they fail completely - leading to no-start conditions. The way to check them is to remove them from the carburator and put a very short (and same thread) cap screw in place of them (with a sealing washer at the head of the bolt of course). If the problem "disappears", then you know you need a new fuel shut-off solenoid. They usually fail from heat, and so sometimes they will work for a little while until the engine warms them up a bit. I know this, because I've had to deal with the failure on more than one occassion when I used to work on Lawn & Garden equipment for a profession. Additionally, you want to make sure you replace the bad fuel shut-off solenoid with a new one, as it's what helps to keep the engine from backfiring when shut down at various different engine RPM's. Peace out!

I would replace the capacitor if I was you, continuing to do that to your fan can burn out the motor eventually. Note: I cannot edit what I posted a couple posts above (pre houzz merge), what I said about the capacitor was wrong. The capacitor charges constantly when supplied power, it just discharges when the fan is turned on, giving the boost needed to make the fan move from stand still. After that, the fan motor keeps the fan moving. It wouldnt hurt to give it a minute to charge before using after replacing with new. In reality it probably only takes like 10 to 15 seconds. That is what I was going for with that comment. Probably the time walking from the panel to the switch is long enough.