See this original post for why I am fundamentally and technically bearish on the US Dollar for the long term: Thoughts on the US Dollar, Analysis of the USDX Long Term, Follow up on the Gold Blog - Sep 17, 09. And per my Dollar/Equity correlation charts, this does not make me "bullish" on equities. What this is illustrating is a loss of confidence in US Sovereign debt, the US currency and the US financial markets at the same time.
My contention is that this can absolutely all happen at the same time. And the question is not one of simply "inflation or deflation".
I have been hammering on the topic because it is the critical issue to understand how all of the advanced economies government's action will not only fail to produce the desired effects, but will more importantly make matters worse. The main issue is Debt Saturation - http://caps.fool.com/Blogs/ViewPost.aspx?bpid=357428. It is critical to understand that an increasing debt load has decreasing marginal utility and there comes a point due to servicing requirements that all new debt has a negative economic impact. This is why we were NEVER going to be able to borrow and spend our way out of a crisis that was caused by too much debt to begin with.
This sets up an extreme deflationary environment (this debt load is unsustainable) within which the Federal Reserve will monetize unprecedented amounts of debt at unprecedented rates. Which will result in a simultaneous deflationary and inflationary outcome: stagflation. There is NEVER anything in economics and especially macroeconomics that has only one cause and one effect. There are always multiple effects with varying degrees of influence (both in absolute value and transience). There will be deflationary impulses and there will be extreme monetary inflation, the Fed will see to that. Which means that I think the most likely outcome will be a combination of the two: stagflation. Economically correlated assets go down in value (like your home and equities as a general asset class) and things you need to buy/consume (such as real assets / commodities) cost more. Really the worst of all possible outcomes.
I do think that most inflationists discount the amount of debt that is collapsing (even though most deflationists use measures like M2 and M3, which have a lot of non-monetary components to prove their point) while at the same time most deflationists discount the amount of monetary inflation the Fed can generate (they argue that the Fed creating base money is like pushing on a string because the banks don't have to lend, even though I am many others have pointed out that the Fed has gone around the banking system and has started monetizing private sector debt directly, which is a trend that is likely to increase not decrease). Most people on either side of the debate is not considering strong evidence that both forces are significant.
So in short, I not only think it is possible that we have falling equities and a falling US Dollar at the same time, I think it is likely.