Comparing points in a bivariate space I have bivariate data from which I have generated thousands of bootstrapped estimates within each of two conditions (pink & blue):

I'd like to determine whether these conditions' bivariate distributions have different central tendencies. 
If I were dealing with univariate data, I'd compute, within each point, the .025 and .975 quantiles of the bootstrapped estimates for that point to construct a 95% confidence interval then compare the intervals of the conditions. Indeed, that's what the lines represent in the above graphic. However, I feel that comparing the conditions on each dimension separately ignores the fundamental bivariate nature of the data, yet I don't know what the appropriate procedure is for bivariate data.
Note that any suggested solution should rely on just the bootstrapped estimates and not the raw data. This is because the estimates in this particular case actually come from rather complicated models that attempt to take into account and remove differences between the conditions that are present in the raw data.
 A: Under the assumptions that both your samples are MV Gaussian with fixed variance (i.e. you are only interested in the differences in central tendency), then you ought to use Hotelling's two-sample T-square statistic. 
It's easy to implement in $\verb+R+$, tough you can find it in package $\verb+rrcov+$ provided you set the $\verb+method+$ option of $\verb+c+$

A: If you accept bivariate normality of the bootstrap estimates, I believe you could do the following:
First, you fit a bivariate normal model to all the data (a model that supports no difference between the two classes). Next you fit a model that consists of two bivariate normal models conditional on the class (your choice whether you do this homoscedastically or not). This would be a model supporting a difference between the two classes.
Now, the double bivariate model is a supermodel of the simple bivariate normal model, holding, in the homoscedastic case, 2 more parameters (bivariate means in one of the classes) and in the heteroscedastic case, 5 more parameters (here, the bivariate covariance structure is added in one of the classes).
As such, you can use a likelihood ratio test for the need to use the more complex model, i.e. whether there is proof for a difference between the groups.
If bivariate normality is not an option, but you have another distribution you believe to be credible, I think this method should work just the same, although the fitting may be slightly more tricky then.
A: If you had a model for the relationship between the x and y axis scales you might be able to do something.  For example, if you z-transformed them so that each measure had equal impact with respect to their variance and, if they were paired data, then you could could calculate a euclidean distance and then an effect size of that distance and also a CI.
But that's a lot of ifs....
