Now however consider, you advertise as well. If some people instead click on your Adwords, then it may effect your CTR in the SERPS. Overtime if google use a CTR ranking algorithm as a factor in the SERPS, and didn't take into account adwords clicks for the same domain, your site may drop in the SERPS.
However this is all just speculation
However has anyone considered that CTR may also be used in ranking the SERPS.
I've said a few times that I am not an authority regarding the SERPS, and that is still true. However, I will say that in nearly three years here at Google, I have never heard anything at all to indicate that CTR is monitored for the SERPS.
That'd certainly be a new one on me. ;)
AWA
Imagine looking at your search engine results for Widgets, and if say the CTR of the 5th result was ten times higher than for the first, than, maybe the 5th result should be higher.
It would be like coming full circle. Taking some of the magic that make Adwords so successful, and coming full circle, and using some of the same ideas in the SERPS.
In the natural SERPs CTR would only encourage spammy people to have more compelling sounding listings, regardless of what content the surfer arrived at.
We're in a similar position to yourself - but we use the same domain. Our aims are to make sure that we appear in all the various country specific Google engines and also on its various partner sites (including those that only show adwords). Only with adwords can we achieve this.
I think that there are also some users who click on adwords first as adwords can be more targeted/relevant to what they are looking for than normal search results.
Using the one domain for both is fine - it simply gives the user a choice on where to click.
We only ever use the one domain for both listings and adwords and have not encountered any problems - but we focus a lot on branding. The best solution for you depends very much on your site/market.
An easy solution would be to use two adwords for your ad group - one with your normal domain and the other with the alternative domain. By doing this (and using a bit of tracking) you should be able to get an idea of which domain gets the most click-throughs/conversions for you.