Dear Luciano,

I would suggest that whatever is causing the bimodality in your scatter plot should be identified and included as a separate variable in your model. You could then consider a suitable technique with two independent variables (one scale and one binary) - either ANCOVA or multiple linear regression.

What is yout response variable (Y)? Is this possibly a binary variable (coded for instance by 0 and 1), or a proportion (restricted ot the interval [0,1]) or something similar?

From an information-theoretic point of view:

The bimodality of the data tells you that there should be a predictor speparating "higher" and "lower" values. One would for instance find bimodal data if one had body size data from humans. This was possibly obtained from an experiment looking for the impact of, say, age on body size. Looking at the data may already show a bimodal distribution, but after adjusting for age (fit a model with age as predictor) the residuals will clearly show a bimodal distribution. This tells us that there is a predictor missing that would explain this separation. Here it is simple: we known it's the sex, since males are often taller then females. Including this predictor will (considerably) remove the bimodality. This was a toy example with clear, simple and known relationships. Real-world problems may be more difficult to understand, but the principle is the same: strange patterns in the residuals tell us that the model is somehow be mis-specified, either structurally (trying to model non-linear relationships as linear etc) or it misses relevant predictors. Bimodality often indicates missing predictors or a missing consideration of interactions.

Given you can't identify what is missing (there is no more information) and apart from this your model specification is sensible (and there are no other similarily sensible options) then it is obviousely not ideal, but the best you can do. If you report the results from the modelling you should also note that problem. The results you present are not wrong, because the results are always based on the assumption that the model is correct. It is always neccesary to reflect if this assumption is reasonable. In your case it can be reasonable in so far as you do not have any better alternative. And publishing the underlying problems can also motivate others to search and possibly identify the missing predictor in future studies.

Bimodal and Normally distributed are contradictory, the distribution can't be both. As implied by at least one other response you may have a mixture of two distributions

you may wish to read this papers.

Article TO DETERMINE SKEWNESS, MEAN AND DEVIATION WITH A NEW APPROAC...

Article PERCENTILE BASED HISTOGRAM BIN WIDTH

If you make a comparison of means between two groups, it is good sign that the data are bimodal! Note that the hypothesis for anova is not that the data are normal, but that the errors are normal, which is not at all the same thing.

On your diagnostic plots, there is no clear sign that residuals (that is, estimated errors) are bimodal. What is clear is that predicted values (that is, predicted data) are.

So without a clear explanation of what you are willing to do, bimodality is not a problem (but you're residuals do not seem to be normal, so you're may be still in trouble...).