Abalone Cove, shown in these five photos, is/was a popular LA cliff-jumping spot, although I’m not sure of its current status. It’s nice for other reasons too. Before the cliff-jumping spot, there’s an area where you can snorkel and see purple sea urchins, kelp, and big orange Garibaldis!
Important Disclaimers: (1) Abalone Cove may no longer be open to the public, so check before going. (2) Abalone Cove may not be safe. People have drowned there (in July 2015, July 2014, maybe more). Because it’s an ocean spot, the water can be rough (and, even if it’s not, conditions can change rapidly), and the depth fluctuates dramatically with the waves. The choppiness and sharp rocks make climbing out difficult, and there’s no good/easy spot to do it. (3) There’s a big rock in the middle, so only jump when the waves and the tide are high. You can find tide charts online. (4) Never jump in blindly. Before jumping, no matter how cold the water is, no matter how many times you’ve jumped there before, and even if you see other people jumping and swearing that it’s “totally fine,” always enter the water *safely* and swim around to check the depth and to make sure there are no (unexpected) rocks, logs, concrete, rebar, or debris of any sort. Don’t put your own body/life at risk and don’t put at risk the bodies/lives of bystanders who might try to rescue you. Be safe (and don’t litter).