B737 Captain reported a NMAC with another B737 on initial climb after take-off.

Narrative

ZZZ was departing XXL/R. We departed XXL; with instructions to turn left to a heading of 260. Aircraft X had been cleared to take-off from XXR just prior to our take-off clearance; with instructions to turn right to 020. As we were beginning our left turn; I noticed that the right window was filled with an aircraft. Almost immediately the 'DESCEND NOW' command came into my headset; along with the warning box. The aural alert sounded at least twice; and my immediate thought was that a command to descend when I was only at 450-500 AGL was unusual; to say the least. However; I pushed the nose down firmly but not aggressively to silence the warning. Our total descent/altitude loss was probably in the range of 50-100 ft. When we were clear of the conflict; we continued our climb; and the First Officer (FO); as pilot monitoring; reported the RA to the Tower. Aircraft Y also reported an RA to the Tower. I have no idea what might have triggered this. The winds were gusty; but only 2 degrees off the center line of the runway. But I have never had a 'DESCEND NOW' command so close to the ground. Likewise; I don't ever recall seeing another aircraft that close on a departure from ZZZ. It's not uncommon in ZZZ1; but very unusual in ZZZ.