A P46T pilot lost the #1 ADAHRS and while troubleshooting that and other related failures experienced an altitude deviation.

Narrative

On climb-out; I lost the #1 ADAHRS; while troubleshooting that failure I received a TCAS failure warning; Acknowledged the TCAS failure and returned to troubleshooting the ADAHRS failure. Then I received an Alternator failure annunciation. I Acknowledged the alternator failure and did an alternator controller circuit breaker pull and reset; this fixed the Alternator failure. I turned my attention back to the aircraft and noticed that I had climbed through my assigned altitude of FL230. I contacted ATC and advised them of my altitude deviation and reported that I have had an avionics problem. They acknowledged and cleared me to FL260. I acknowledged FL260 and then received a cabin altitude 10K annunciation. I Acknowledged the cabin altitude10K; silenced the horn; and selected a lower altitude on the pressurization controller and requested FL240 from ATC. Monitored pressurization and determined that we were pressurized; just not holding maximum pressure. Asked ATC for FL240 as a final and we had no more problems from the pressurization. I turned my attention back to the ADAHRS. I was finally able to get the ADAHRS to come back online; but by that time it had lost its initialization and would be useless the rest of the flight. I was able to restore function to all failed components except the ADAHRS. I advised ATC of my reduced operational capacity and continued the flight with no further problems.