It's a blazingly warm fall day as we load our scuba gear into a small boat at the Cortez Club, the epicenter of the La Paz diving and snorkeling scene. I'm about to head out into the Sea of Cortez in search of hammerhead sharks, that elusive creature I've been determined to come face to face with since my first dive experiences more than twenty years ago, and I'm trying to think positively. After all, this is the place that Jacques Cousteau – the Frenchman who basically invented scuba diving and was the custodian of the world's oceans for most of the twentieth century – called "the world's aquarium," and hammerheads are known to love the warm waters.

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But I've been teased with hammerheads in other wondrous underwater destinations. I've strapped on my tank in Fiji (coincidentally very close to where Cousteau's son Jean-Michel built his luxury resort in Savusavu) and had the strangest dive of my life, swimming with seven other divers out into the open ocean on the search. Crystal-clear water above, below, beside – complete nothingness surrounded us. Only the sound of my own breathing and no coral, no tropical fish, just water, water everywhere. Completely surreal, and it went on and on and on, a 50-minute underwater swim that became a Zen experience, but with not a hammerhead in sight.

I tried in the Galapagos, too, and never saw a single hammerhead, on a couple of attempts while visiting that bucket-list spot. So now in La Paz, as we pull out into the sea, I'm thinking I should have started here, as it's such an easy place for an Angeleno like me to get to. Two hours on a plane from LAX to Los Cabos plus an easy drive north, I took a stop in Todos Santos for fish tacos and a frozen drink at the original Hotel California before rolling into La Paz, with its lovely malecon waterfront esplanade and azure bay. It's nothing like nearby Cabo San Lucas with its massive resorts and big crowds slurping margaritas as they can get them; instead, La Paz is the place for water adventurers, people more interested in diving, snorkeling, kayaking, stand up paddle boarding and deep-sea fishing. Plus, there are still plenty of fish tacos and margaritas.

But back to that boat. I'm soaking up a bit of vitamin D as our vessel leaves the shelter of the bay and begins to slam through waves, rattling the dive tanks that line the back end near the motor. Choppy seas, never a good sign, I think. And I'm right. Forty-five minutes into the hour-plus trip to where the hammerheads hang out, our captain slows to a crawl and announces we're not going to make it, the sea is too choppy. I try not to sob out loud, but a sad groan does escape my lips.

The day was still perfection, blue skies and, at 100F, maybe even a tad too hot for me. We still got in two dives, one to the Fang Ming, a cargo shipwreck that offered up a huge sea turtle who hung around for the whole dive. Dolphins danced around the boat on the way back to shore, too, so it wasn't exactly a lost day, but still, no hammerheads. Sigh.

Meanwhile, my friends had left the Cortez Club that morning, too, on a different ship that went on the hunt for another incredible ocean-living creature, the whale shark. Whale sharks are plankton-eating sharks (not whales). They're massive, gentle fish who migrate into the Sea of Cortez every October and stay until early spring, usually in the area called the Magote, just a short ride from the La Paz shoreline. Protected by the Mexican government, only snorkeling and free diving is allowed near the whale sharks, who can grow up to 40 feet long and weigh nearly 21 tons.

They, of course, not only found one whale shark that day; they found six. Which made my next day's choice even harder to make. Once again ready at the Cortez Club with all my diving gear in hand at 8 a.m., once again a sunny, gorgeous day in this desert paradise, what was I going to do? Whale sharks, which seemed pretty much a guarantee, but which I had experienced near Cozumel a few years before? Or try yet again for those damn elusive hammerheads?

Hammerheads. It had to be hammerheads. I'm determined to add them to my diving checklist, which already includes sharing the water with a huge list of sharks, manta rays, moray eels, lionfish, octopus, squid and the hugest stingray on the planet. So off we go again, five hardy divers and our local dive master, who reassured me that today was the day. "It better be," I thought, "since this is my last chance on this trip!"

This time, we made it out to the right spot, but then things get pretty hairy. The captain and the dive master struggle to find a spot to let down an anchor, we're bobbing around in pretty rough water and hanging on for dear life. But finally, they settle the boat and we drop into the water, grabbing the anchor line to avoid being swept away in the hard-driving current. Down we go, to about 60 feet to the sand, where we're greeted by five huge Moray eels, all poking their huge heads out of a coral formation, chomping their massive, razor-sharp teeth in the air, looking for lunch. That alone would have made this Sea of Cortez dive unforgettable, but the current pulled us away quickly – toward where the hammerheads hang out.

As we slide across the water, the sand drops away, the sea floor now more than 200 feet below us. The visibility begins to get murky. We're peering through the churned-up waters, barely having to swim at all as the drift dive progresses. Is there something out there?

And suddenly, there is! Through the murk, the dive master sees two hammerheads and points excitedly at them. They are close, with those distinctive rectangular heads turning for a quick glance at us, then gone. We swivel our heads, hoping for a glimpse of more, but that's it. My quest is finally complete – but it was such a fast encounter I didn't even snap a single photo.

In fact, that little taste of those majestic sharks has just wound me up; now I want more, in clearer water. So it's a good thing La Paz is such an easy destination to reach from Los Angeles, for I'm going to have to go back soon and try again. Perhaps in January or February, when the gray whales come to have their babies in those warm waters and I can have an up-close encounter (inside a boat) with them, too.