LR-60 Captain reported a pressurization malfunction along with a navigation compass system malfunction during climb. The crew diverted to an alternate airport and landed.

Narrative

After what seemed like a normal takeoff into almost immediate IFR conditions; we realized the aircraft was not pressurizing and soon thereafter; the amber emer press light illuminated.We complied with the QRH emer press checklist to no avail while the passengers were loudly complaining of hot air in the cabin at which point; since we were already not pressurizing; we turned off the bleed air switches which stopped the flow of hot air into the cabin.At this point; to make matters worse; the amber 'H' light on the pilot's primary display illuminated signifying we had a problem with our nav compass system which was confirmed by ATC questioning our heading.We then complied with the QRH checklist for the 'H' light which did not solve the heading discrepancy.We then [requested priority handling] because of our unreliable compass system; IFR conditions throughout the entire ZZZ area; instrument approaches to near minimums at the departure airport; not to mention being considerably overweight for landing.We then requested clearance to a suitable VFR condition airport; ZZZ1. Enroute to ZZZ1 at 9000 feet at ATC's request; the compass malfunction rectified itself; we burned off the required fuel to land below the Maximum Landing Weight and the landing was normal and uneventful.