Are air purifiers and ionizers effective?
Before I bought an air purifier a few years back, I found a Consumer Reports test of such machines, including several filter-based models and at least one ionizer.  The ionizer received a failing grade.  According to Wikipedia, the manufacturer subsequently sued Consumer Reports for product defamation and lost.  I don't have the Consumer Reports article anymore, but if I remember correctly, the only effective air purifiers were filter-based. Only a few models had much effect, and only when the fan was run at the highest (and loudest) speed.

We have had a filter-based air purifier (made by Sears) for about three years.  It cost about $300, and the annual filter cost is about $150.  It is effective in the sense that the filter does get dirty, but my (subjective) impression is that its cleaning effect is small compared to regular vacuum cleaning and dusting.  We haven't noticed any health improvements—but we didn't have dust allergies or other dust-related health problems either.

Apparently there have been studies into high levels of negative ions as a treatment for seasonal-affective disorder.  I have not read the papers or otherwise evaluated the research, and I don't know if commercially available air purifiers produce anions at the levels that are claimed to be clinically relevant.