Captain reported baggage compartment door was not secured correctly and departed the aircraft in flight. The flight crew continued to destination airport where the missing door was confirmed.

Narrative

We were scheduled to fly from ZZZ to ZZZ1 with a XB:00 departure time. It was windy and raining moderately at ZZZ as we approached departure time with a line of thunderstorms west moving east-northeast showing lightning on the horizon. It was still quite dark at XA:40 when we closed the door to do the cockpit prep and pre-flight checklist.I had performed the walk around except for loading our crew bags in the baggage bin. I told my Co-pilot to load those up and get into the airplane and work on a clearance by phone while I moved and parked the rental car from which we had just unloaded our crew bags.As I walked back to the airplane I glanced at the baggage door just to make sure it had been closed. This was done from probably about 25-30 feet; essentially looking for a 'black hole' to indicate the door was not closed.We taxied out to Runway XX and received an initial departure clearance for a turn to 050 and an initial altitude of 3;000 feet. There was moderate rain and the wind was gusting from the west-southwest to west as the line of thunderstorms was getting closer.Takeoff was uneventful; with no indication of any doors not being closed. I had delayed rotation slightly because of possible wind shear from gusts from the thunderstorms. We were lightweight and with cool temperatures the aircraft accelerated quite rapidly. At approximately 300 feet to 400 feet AGL; just as we were entering the overcast; we encountered a brief period of moderate turbulence. It was at that time the 'Door Open' Master Caution Light illuminated and showing an 'ACC' (Accessory) Location Light which meant the baggage door was not seated properly. There was no noticeable vibration or unusual sound associated with the caution light coming on. I reduced power to preclude an acceleration to normal 250 KIAS and climbed at about 210 KIAS which was the approximate speed when the door opened. Due to the deteriorating weather conditions at ZZZ with low ceilings; rain; and gusty/strong winds and with no flight control issues I decided a return to ZZZ was not a good choice and elected to proceed to ZZZ1 at lower than normal speeds. After maybe 30 seconds or so with the Warning Light on I heard a loud bang and commented to the Copilot that I hoped that wasn't the door departing the aircraft. We continued on to ZZZ1 and landed without further incident. I did not declare an emergency as there were no issues with aircraft handling or proper power. Upon shutting down at ZZZ1; the Copilot went immediately to check the door and reported it had departed the aircraft along with a canvas style flight bag of his. We secured the aircraft and asked the FBO to place the aircraft in a hangar to preclude precipitation from entering the baggage compartment. In the aftermath; I contacted the Aircraft Owner and the Airport Manager at ZZZ. I asked the Manager if anyone had reported an object falling and asked him to contact the police department to find if there were any such reports. None had been made and I left my name and phone number in the event anything was reported. I then called the ZZZ FSDO to report the incident; and took photos of the areas of damage.I believe the door was closed by the First Officer (FO) with the leading edge hanging strap wedged between the door and the door frame. This allows the latch to close and will appear closed and latched from the exterior; but the latch will not be secure because it has not engaged the post to hold it shut. The turbulence likely exacerbated the opening enough for the door to open itself into the slipstream and it separated from the aircraft. I have done the same thing ... it's quite easy and very common. But I always visually check the edges of the door with adjacent fuselage to ensure full closure. In the dark and in the rain; I believe the First Officer missed the problem and I did not physically check it myself.I could have re accomplished the door closing but it's not realistic to have a two pilot crew and rely on the Captain to physically re-check every action by the co-pilot. I've had the strap get caught and the latch appears to be closed but isn't actually latched. I believe there could be a bungee cord attached to the baggage door straps that would not allow the straps to remain between the baggage door frame and the fuselage.