Well, I guess I better be an economist here for a little bit. In hitting that stagflation link, I couldn't help notice that there was not one mention of the money supply in all the discussion of "inflation." As is often the case, the media confuse "increasing prices in several sectors due to higher oil prices" with genuine inflation. If oil prices are going up, that's certainly going to cause some prices to rise. And it wouldn't be surprising, with this good now relatively more scarce, that growth would suffer as well. But that is not stagflation of the sort we saw in the 1970s, where the Fed was jacking up the money supply and genuinely causing across-the-board inflation. It's a simple point but increasing prices of some goods due to a scarcity (whether real or contrived) is not inflation, thus the stalling out of growth that results is not stagflation. The only thing that deserves to be called inflation is a true across-the-board increase in prices which can result, with only very few highly idiosyncratic exceptions, from an excess supply of money.
In addition, note the subtext: the whole term "stagflation" emerged because people thought, for a few decades, that higher rates of inflation would reduce unemployment and increase growth (what economists called the "Phillips Curve"). If true, this would make the notion of simultaneous high inflation and high unemployment to be a puzzle. When it happened in the early 70s, the financial press had to invent a word for it - "stagflation."
Of course the Phillips Curve trade-off was an illusion from the start, as Friedman, Phelps and others demonstrated theoretically and as the 70s showed empirically. Other economists had long argued that inflation in and of itself reduced growth, so that the notion of high inflation and low growth was completely comprehensible and to be expected. We don't need a word for it, rather we just need to better understand inflation. (I should note that the "costs of inflation" are a particular interest of mine - see my article "The Costs of Inflation Revisited" in the March 2003 issue of The Review of Austrian Economics, or the chapter on inflation in my 2000 book.)
Bottom line: I'm not worried about "stagflation" until I see the relevant money supply figures.
Now, as for which candidate would do a better job on monetary policy... good question. Do keep in mind that whomever is running the Fed, the biggest constraint on the Fed's behavior these days is the speed and ease of international financial transactions. The Fed simply can't afford to inflate because people can leave the dollar much more quickly and easily than 30 years ago. International competition has a great deal to do with the reduction in the US inflation rate (although Greenspan deserves credit, as does his predecessor), and that competition will face any new Fed chair regardless of who is president. There is a lot more consensus in the economics profession about the real costs of inflation, or at least its inability to produce any real benefits, than there was 30 years ago, so I'm less concerned about it than earlier.
I will give one caveat, though: as the deficits and debt continue to grow, the benefits from inflation to the central bank begin to grow, particularly where central banks are not so independent. It's not out of the question that continued high deficits in the US would up the political pressure on the Fed to monetize some of that debt. And that would give us inflation and, likely, stagnant growth.