I've done both, a couple times. Two weeks notice is always a bad idea, and a generous leave notice is always favorable to future employers. It's a pretty good sign that the dude isn't too worried about HIS situation or their's.
1) I left with two weeks notice, taking an international job, so there wasn't much wiggle room. It sucked, and I kept good rapport to this day assisting them as much as I could by building documentation and working with my team for those two weeks and random calls thereafter. The team replaced me.
2) Left the international job, gave them 7 months. Took 5 to find the new guy. Politics and teaching him where to find information were the killers. He shadowed me for a few weeks, then we started splitting assignments, and then I took a mutually accepted garden leave for the last two weeks handling only critical tasks that shouldn't carry forward. Rapport is strong with these companies still as well.
3) Started the new job. Hired on Friday, started Monday, didn't even have a place to live. This one was a mess and I HAD to start on Monday because the bugger was GTFO. No documentation or best practices. Went through every server, what OS it was running, what applications, and the access. Started to canvas the network, only to realize one week wasn't enough. Focused on access and design realizing quickly it was a big ball of duct tape with VLANs and different OSPF styles.
When it really came down to it, I gauged my skillset, supplemented my tools and information with whatever could carry the biggest impact, and prepared myself to crash and burn. Now, less than a year later, I have started rolling out best practices based on my documentation and there's been a stark change as the enhancements and simplification has begun winning the battle. But it was a big uphill battle. Your armor will be more important than your weapons for a while, but every chance you get to use your weapons to correct something for win-win, do it and don't look back.