ANCOVA versus Multilevel Models with baseline and follow-up responses
(Some very useful advice from Thom Baguley)
The ANCOVA versus multiple model argument can get confusing. First you can have covariates in either multiple regression or multilevel models so including baseline as a covariate is a legitimate option either scenarios. The main alternative is an ANOVA style model with 2 groups and three repeated measures. This could be run as a regression or multilevel model so the two issues: 1) baseline as covariate or ANOVA-style model, and 2) single level regression or multilevel regression model are not tied together.
1) This issue comes up in the literature as Lord's paradox. In essence the two approaches test different hypotheses and usually the covariate approach is more appropriate as it relies on less stringent assumptions about the impact of the baseline score on the outcome Y. Adding baseline as a covariate will generally be a better approach and have greater statistical power. I summarise the arguments in my book Serious Stats (pp 652-4) but other good summaries exist - notably by Dan Wright and Stephen Senn. If adding baseline as covariate it helps interpretation to centre the covariate and analyse the change scores (Y - post test) and (Y - minus follow up).
More generally adding a between-subjects covariate to a repeated measures model doesn't usually help much because the if omitted it just gets sucked up by the subjects term (which is separate from error). However it can make a difference if you add covariate by condition interactions or if you have a time-varying covariate.
2) If the design is balanced with fixed factors and the assumptions of an ANOVA are met there is no advantage to running a multilevel model. In more complex designs a multilevel model has advantages in declining with imbalance, being able to handle additional random effects, time varying covariates and relaxing assumptions about the form of the covariance matrix (sphericity or multisample sphericity). You can also generalise the model to handle discrete data.
With two time points and just one covariate (baseline) I can't see much reason to use a multilevel model. Sphericity can't be violated in a model with only two repeated measures. If you have missing observations or some other complicating circumstance. However you haven't provide much information.
For example, if you are planning to analyses totals or means from multiple trials in the ANCOVA then the multilevel model would allow you to model the trials nested within participants. This sort of model would be quite a bit more complex but might be attractive in allowing covariates that vary between trials (and most likely having higher statistical power).