Very polar amines can also be purified by ion exchange chromatography.
If the reaction is clean/does not contain other amines as impurities, you can simply load the reaction mixture onto a short column of a strongly acidic resin (I used to use Dowex 50W X8 for this regularly). The amine is protonated and immobilised on the resin, you can then wash off salts etc with water, and finally liberate the amine by flushing with dilute aqueous ammonia. If you have a mixture of amines you can use various resins to separate amines in a more rigorous chromatographic purification (e.g. Amberlite, phosphonics cartridges etc.). There is also reversed phase chromatography as well.