NEW YORK – New York is a city in constant flux. Today’s corner deli is tomorrow’s Starbucks and, the day after that, you find a high-rise with big shiny letters spelling “TRUMP.”

The Manhattan sidewalk is packed not only with people absorbed in that moment, but also with ghosts from the past. Anyone, it seems, might have crossed your path before. Alexander Hamilton? The Olsen twins?

But what if you had been around, year after year, to experience it firsthand? John Amsterdam has. That’s the premise of the new Fox drama “New Amsterdam.”

Even without Botox, he looks maybe 35. But John has lived on this island since the 1640s, when, as a Dutch soldier, he saved the life of a Native American woman. In return, she rewarded him with immortality.

Ever since, the citizens surrounding him have come and gone. But John is stuck, fated to be left behind by each generation – until he meets his One True Love. Only then will the spell be broken. He will join her and grow old. And he’s ready. After all this time, he wants to get on with his life.

Premiering 9 p.m. EST Tuesday (with a second episode 9 p.m. EST Thursday), “New Amsterdam” is a mystical romance. It also is a time-spanning fantasy – and a contemporary police procedural: John’s present-day job is that of an NYPD homicide detective. A man who might never die solves murders for a living.

“When I heard about a show with an immortal detective, I was like, ‘Ummm, I’m not sure,'” says Nikolaj Coster-Waldau (“Kingdom of Heaven”) from his home in Denmark. But after reconsidering, he now stars as John.

“Once you get through the crazy setup,” he explains, “you realize the character faces the same problems we all face – he’s just been facing them a long, long time.”

The first episode of “New Amsterdam” is effective in establishing this “crazy” (and rather ambitious) premise. (Granted, it’s a premise that bears similarity to Pete Hamill’s best-selling 2002 novel, “Forever,” as Hamill has complained. The series’ executive producer David Manson has countered he was unaware of the book’s existence until production on the show wrapped.)

But the premiere’s success is all the more impressive to anyone who got a peek at the original version of this pilot, which was sent to critics last summer. A then-and-now comparison reveals a promising concept that, in its first rendition, had potentially fatal flaws.

“New Amsterdam” 2.0 deftly fixes those flaws, and gets the series off to a strong start, thanks to several scenes reshot, a minor role recast, some strategic re-editing, and – underlying all the tweaks – a heightened sense on the part of the producers for what the series is about. Right now the pilot system is losing favor among budget-slashing networks. But the pilot for “New Amsterdam” served a valuable purpose: it was a good way to find the weak spots.

Shot on location last March, the initial version is “what I euphemistically call ‘the selling document,'” says Manson (whose past series include TNT’s “Saved” and FX’s “Thief”).

“But I think the network realized there was still work that needed to be done on it” once the series was ordered, adds Manson, speaking from L.A. “Fortunately, Fox gave us the resources to go out and do the necessary reshooting and reconceiving.”

The biggest modification: Originally, John Amsterdam was bitter and depressed about his immortality.

He happens to be a recovering alcoholic. He has been sober 15,495 days, he declares at an AA meeting. But then he got overwrought after having possibly crossed paths with The One while on a crowded subway platform. He felt compelled to have a drink.

This crucial scene took place at the bar owned by Omar (Stephen McKinley-Henderson), Amsterdam’s loyal confidant.

“If I can find her, time will have value,” he told Omar, insisting that’s “a good thing.” Yet he seemed pretty desperate while saying so.

“He’s waited 400 years for this moment and he finally has it, and then he decides to lose his sobriety over it? That was confusing to the viewer,” says Manson.

In the bar scene as re-shot, Amsterdam looks happy instead.

“This is like nothing I’ve ever felt,” he tells Omar. Somewhere near him on that subway platform, “she was there,” he says. “I know she exists!”

So do viewers, who saw his apparent soul mate: a lovely young woman named Dr. Sara Dillane (Alexie Gilmore).

This time, John feels no need for a bender.

Coster-Waldau says playing a cheerier Amsterdam “was a big change for me. I had to redo my analysis of the character” when reshooting the scenes last summer. “But it made sense: He should be a character of hope, not disillusionment.”

Besides brightening the hero’s mood overall, other changes were made for the New “New Amsterdam.” John’s partner, Eva Marquez (Zuleikha Robinson), is introduced much earlier in the revised episode. And right away Amsterdam identifies himself to the audience as a cop, whereas, in the initial version, he didn’t spell out his profession for almost five minutes, while, instead, the action dwelled on him in his 17th-century persona.

“We didn’t want the viewer to wonder for too long who this guy is,” says Manson.

Who is he? A 21st-century guy who knows better than any New Yorker that life goes on – but, so far, not for him.