B737 Captain reported Master Caution warning and hydraulic system malfunction after retracting landing gear. Crew diverted to an alternate airport and landed overweight.

Narrative

After departure upon retracting the gear; the master caution light illuminated with low pressure indications for both system A" engine and electrical pumps. Electrical pump light extinguished shortly after illuminating but engine pump light remained illuminated. We focused on flying the departure procedure as it is slightly complex out of ZZZZ due terrain as well as moderate turbulence on this day. Once we cleared terrain we noticed that system "A" hydraulic quantity was at 76%. I contacted dispatch and discussed the issue with Maintenance Control on the line and we agreed that a diversion was necessary. I elected to divert to ZZZZ1. Hydraulic pressure en route slowly decreased from 76% feet 72%. I was task saturated for nearly the entire flight between the maintenance issue; the complex departure procedure; the moderate chop and turbulence; conversation with dispatch and Maintenance Control and thinking through the diversion and possible outcomes from the situation that I did not declare "pan; pan; pan" with ATC. I simply requested a left hand turn and a diversion to ZZZZ1 in the moment which ATC provided. This was during the conference call with dispatch/ATC while the FO was flying and working the radios. Everything else on the flight I thought went well considering the circumstances. We attempted to burn enough fuel to land overweight but landed about 500 pounds overweight. I decided this was a better option than attempting to spend more time airborne to get under maximum landing weight."