Cessna 680A flight crew reported while being vectored for approach in IMC they got several CAS messages. Engine control faults for both engines; both attitude and heading and air data computer faults.

Narrative

On arrival into ZZZ; during vectors onto the RNAV Y XX approach and at approximately 3;000 ft; we experienced numerous anomalies. The primary CAS messages were amber ENG CNTRL FAULT for both left and right engines; amber BOTH ON ATT/HDG1on both Primary Flight Displays (PFD's); and BOTH ON ADC2 on both PFD's.I do not recall the frequency we were on at the initiation of the event; but it was the last approach frequency prior to being transferred to ZZZ Tower Control.When the CAS messages displayed; the Autopilot and auto-throttle systems failed; and the magenta flight director began flickering on and off and jumping around a bit; so we turned off the Flight Director (FD); requested vectors off of the approach for time to diagnose our issues; and requested a higher altitude. Once established on vectors; we positively transferred controls and radio to the second in command (SIC) and I began running QRH's for the displayed CAS messages. The QRH procedures did not succeed in restoring the malfunctioning systems. We briefly discussed the new company policy of contacting the chief pilot in instances like this; but considering the approaching storms; potential time compression to beat them into ZZZ if we delayed too much longer; the prevalence of maintenance personnel available at ZZZ already; the fact that we were already well below 10;000' so wifi was not available to communicate quickly and effectively; and we were only about five minutes from the airport; we opted to continue without making that contact. (After the flight; I called the chief pilot to discuss our decision and he said he believed that we had made the correct decision to just continue to ZZZ without contact due to all the same factors just listed. I also asked if this event required a report; or if he would like one; as the event didn't quite match any of the defined events listed in the FOM that require reports; but was moderately related to a couple of them. He said that we did not need to file one.)We discussed whether or not requesting priority handling was necessary during the event; but opted not to. Our thought for not doing so was that the FADEC's reduced automation; but not the ability of the engines to function; the FD flickering did not effect the LOC/GS CDI on green needles; all of our basic 'six pack' instrumentation remained fully functional; and while we had partially lost the use of our #1 Comm; we retained full use of the #2 Comm and so maintained complete functionality of all our basic necessities. With that in mind; we set up for and requested vectors for the ILS XX and hand flew the approach uneventfully. Weather was listing the ceiling as 900' overcast at the time and that was pretty close to what we experienced; too.The following is the maintenance write up I submitted; but I do have one addition here. Upon reviewing one of the videos I noted that I forgot to include the Oil Temp and Oil Pressure readings flickering on and off. Otherwise; I believe we captured pretty much all of the anomalies that we noticed during the event itself. Here is the Maintenance write up: During arrival into ZZZ and shortly before starting our approach; we received the ENG CNTRL FAULT L-R amber CAS message. Again; both left and right engines simultaneously displayed fault messages. Coincidentally and subsequently; we also received a cascade of other messages and anomalies; which I will attempt to list here:(1) Amber BOTH ON ATT/HDG1 messages on both PFD's; (2) Amber BOTH ON ADC2 messages on both PFD's; (3) Random yellow X over AOA indicators on both PFD's; (4) The magenta flight director flickered on and off on both PFD's; whether on NAV; HDG; or Roll modes; (5) White FD Fail CAS message; (6) White AP Fail CAS message; (7) Random flickering on and off of amber CRU messages on the engine display; (8) N1 V Bug randomly turned amber; (9)(a) while on the ground; the engine N1 indicator outline (the normally green circular outline) flickered back and forth from green to white; (9)(b). While on the ground; the N1 mode limit annunciation flickered from white TO to yellow dashes; (10) Comm 1 active and standby frequencies on GTC #'s 3 and 4 displayed yellow X's and could not be changed; but the last frequency remained active; audible; and the volume could be adjusted; (11) Once on the ground; the APU startup test did not illuminate the APU FIRE switch as it should; (12) With the APU switch turned into the on position; the APU status display randomly flickered between numerical values and amber dashed lines; (13) raw data CDI (I.e. -- green needles for ILS) functioned normally.'Suggestions: At this time; I do not think I would change anything regarding how we handled this event. I will note that; initially; the situation was not so clear. We had a whole lot of anomalous indications in the flight deck to process; and they did not all immediately seem to have anything to do with the FADEC's. I still can't figure out how some of them are connected. Partial lost comm; flickering FD; etc. Even with the items that were related; the sheer volume of flickering indications took a while to process and mentally confirm that they were reasonably explained our CAS messages. (I have no clue what was up with the Comm #1 failing at the same instant; though.) To complicate the matter; my partner's first concern was that we may have been experiencing a bad fuel load; and so there was an understandable mild anxiety on his part; particularly given that we were IMC at the time. In light of all of this; the extra fuel we had onboard gave us ample time to troubleshoot; consider; discuss; and brief our response without the added pressure of trying to beat fuel starvation to the ground. Had we had the new standard of 600-700 lbs extra for this leg; fuel would have; without a doubt; been front and center on our minds; and it would absolutely have effected our ability to address the situation to our comfort. We were burning approximately 900-1;000 pph; and 600-700 lbs 'normal' extra would have maybe given us 20 mins before we were looking at requesting priority handling. In our case; it took us about 30 minutes to break off; get established; transfer controls; run the QRH's; consider the abundance of 'wrong' indications to make sure we weren't missing anything critical; ensure we were set up for a hand flown approach in IMC; transfer controls back; and then return to the airport and land. It was greatly appreciated to have the time available to do all of that at a comfortable pace and with the dialogue necessary for both crew members to assess the situation without feeling rushed. I truly and genuinely understand the fuel issue. I really do. Yet; for what it is worth; our response would neither have been as comfortable nor as thorough had we arrived at that point with what would now be considered a 'standard' fuel load for this leg.