C550 Captain reported a Master Caution with a L FUEL FILTER BYPASS Annunciator lights illuminated after takeoff. The lights extinguished and reappeared 45 minutes later. A decision was made to divert and make a precautionary landing.

Narrative

During the departure phase of flight prior to selection of climb power; a master caution illuminated in conjunction with a L FUEL FLTR BP annunciation. The annunciators self extinguished within a few seconds.Approximately 45 minutes later in cruise at FL390; approximately 30 nm north of ZZZZZ; both annunciators illuminated again.Per my request; the aircraft controls were exchanged to me. Per the appropriate checklist (summary: land as soon as practical); we requested a diversion to ZZZ1 and informed ATC that we had an abnormal indication. A clearance was obtained to proceed direct ZZZ1 and to descend to FL370.A descent to FL370 was made and the thrust levers were retarded. The power on both engines reduced accordingly and the annunciators extinguished. We leveled at FL370. At the new selected cruise power setting; the annunciators remained extinguished.I decided to continue along our previously assigned flight plan from this position.Approximately 5 minutes later; the annunciators reappeared. Engine power was reduced and the annunciators extinguished. At this point; it was clear that diverting was the appropriate course of action.Due to the distance traveled south in that period of time; and the knowledge of it as a suitable location for maintenance; we requested to divert to ZZZ2 and were cleared to ZZZ2 via direct ZZZ3; ZZZZ 7 arrival. As engine power was reduced; the annunciators extinguished.To level at the assigned altitude of 11000 ft.; power for both engines was increased. Subsequently l *and* R FUEL FLTR annunciators illuminated in conjunction with a master caution. At this location; approximately 5 nm northwest of ZZZ3; a decision was made to divert to ZZZ4 which was only approximately 10 nm away. From that altitude and location; the aircraft would have been able to glide to the runway.An idle power descent to a landing was made with all of the aforementioned annunciators extinguished. Upon clearing the runway; both engines were run-up to approximately 70% n1 and all of the aforementioned annunciators illuminated. Upon engine power reduction; all of the aforementioned annunciators extinguished.A request for priority handling was not made during this situation; ATC was very accommodating of our requests and therefore the need to [request priority handling] was not obviated. No further assistance was required.The appropriate checklist for these annunciators does not address a scenario in which the associated annunciators self-extinguish. Additionally; each time the fault presented itself; the master caution illuminated until acknowledged and the FUEL FLTR annunciators flashed until the master caution was acknowledged. This proved to be distracting; especially as dusk set in. For these reasons; a possible corrective action on the part of the manufacturer would be to make the annunciation of a fuel filter bypass a latched fault condition. If it were a latched fault condition; the master caution would illuminate once until acknowledged; the FUEL FLTR annunciator would flash until acknowledged by pressing the master caution button; and then would remain steady-on illuminated. This would reduce the ambiguity faced by the flight crew; make the cockpit annunciations less distracting; and make the pic decision-making process more straight forward.Prior to this flight the aircraft was fueled jet-a positive. After the event; fuel aboard the aircraft was tested for water and def contamination. The amount of water suspended in the fuel was within tolerance of the test. Def was not present in the tests.At the time of writing; the root cause has not been determined and the airplane has not flown again pending additional maintenance tasks including removal of both fuel filters.