Can augmentin (amoxicillin and clavulanate) cause rashes and if so how long will they last after stopping the antibiotic?

Yes. Up to 2 weeks.: Augmentin (amoxicillin and clavulanate) is Amoxicillin augmented with clavulanic acid. It enables Augmentin (amoxicillin and clavulanate) to kill bacteria resistant to amoxicillin. Like Amoxicillin Augmentin (amoxicillin and clavulanate) can cause an allergic rash - hives and angioedema usually appearing right after the 1st dose - or an Amoxicillin rash - red flat or raised spots that don't itch appearing on the 7-10th day of treatment. Once drug is stopped each rash is gone in 2 weeks.

Yes it can: Augmentin (amoxicillin and clavulanate) is a combination of two antibiotics, one of which is ampicillin, a form of penicillin, which can cause rashes that may persist for up to several months, although the rash usually clears before that time.