You can search the web and find a number of PIC based Morse Code projects, but they generally are all based around some tone-decoder external circuitry; some of the external tone decoders costing much more than the PICAXE proper.
This circuit is simple, all the work is done inside the PICAXE in basic software. A single NPN transistor will function as a signal inverter and as a analog clipper giving the input pin a fair square wave to analyze.
Add a Morse Code key and oscillator (555 timer for example) and you have a nice setup to learn and practice. Additionally, you can feed the output of your PC (> 0.8V) earphone jack to the circuit and decode ARRL wave files from 05 WPM to 15 WPM (or greater when Farnsworth is utilized between characters.)