Though alchemy died in its true form hundreds of years ago, what was learned in its study lives on. It started and was based off of religion, but, as it spread, more and more people of different backgrounds began to pick it up and it was decided that to be an alchemist, you didn't necessarily have to believe in it spiritually. As alchemy began to grow more scientific in nature, it began to separate into religion and sciences (chemistry, metallurgy, etc.) and lost much of its mystical aspects. This brought in many more people who believed only in the natural, materialistic aspects. People who openly believed in the mystical powers of alchemy were laughed at or worse, prosecuted for being witches and warlocks. This new perspective eventually led to it dying as a topic of study, but it lives on in science, religion and history.
In modern chemistry, scientists study the way matter works and how particles interact with one another. European alchemy set the basis for this. It started with Aristotle's first idea of four elements and accumulated to Robert Boyle revolutionizing our understanding of matter. Boyle was a great alchemist and through studying Aristotle and his own experiments, he came up with Boyle's law. Boyle's law has to do with the relationship between the volume and pressure of gas. With this new development, Boyle re-introduced the world to the atomic theory first described by Democritus in ancient Greece. Although Boyle probably contributed most to chemistry, almost every alchemist did their part to enhance the field and bring it to where it is today.
Metallurgy is a field of science that specifically focuses on the properties of metal. Alchemy was detrimental to the field ever appearing as it was what started scientists looking at metals. In their search for gold, alchemists discovered a lot about the properties of metal and how they react to one another. It started with the seven principle element gold, silver, iron, mercury, copper, iron, tin and lead. From these seven, alchemists found other metals and used chemicals to create new substances. They eventually learned that these metals were not tied to celestial bodies or the four primal elements, rather they were made up of atoms and molecules that were constantly moving and changing. Eventually scientists moved away from the mystical ideas and outlandish objectives and instead focused on making strong, more functional materials with the alchemical knowledge of old.
Though not as tied to alchemy as chemistry, physics has its place in the alchemical books. With the knowledge we have about particles and matter today, we are able to achieve what alchemists wanted to hundreds of years ago. Through nuclear physics, we have slowly discover the basic particles that make up everything. It's not water, air, fire or earth and it isn't sulfur, slat or mercury as alchemists predicted but the protons, neutrons and electrons are what they were truly looking for. With our knowledge of particle and nuclear physics, we can now change substances to be other things. We now have the transmutation alchemists spent hundreds of years trying to achieve. Though physicists use very different methods than alchemists did, the similarities in their work and results are uncanny.
Medicine and the making of pharmaceutical drugs would not be possible without its origins in alchemy. More so in Chinese than European alchemy, mixtures of different herbs and chemicals created elixirs and potions used to aid people. What started in ancient china as an almost "guess and check" of mixing organic and inorganic material in now a very precise science that some people depend on for their day to day lives. Though some groups of people still rely on herbal medicine, much of the old alchemical work has developed to create important drugs. There are even rumors that large, western companies are researching old Chinese medicinal techniques to develop new drugs.
Though this field is usually regarded as proto science of sorts, it does share similarities to alchemy. Chinese alchemy focused a lot on balancing internal energies and using potions to heal oneself. Today, similar methods can be found in the teachings of massage therapy, chiropractic therapy, reflexology, dieting and the use of "essential oils". Though some people find that these methods do work to heal or aid them, there is little to no scientific proof to back them up. Similar to alchemy, these field are very subjective and yield different results for each person. Though not an exact science, these methods of longevity reflect the search for the elixir of life during the peak of alchemical popularity.
Religion was around far before alchemy, and it is still around long after alchemy has gone. Not every aspect of religion that was present in alchemy survived though. Greek and Egyptian legends are no longer the basis of religions, instead they are mere stories. Christianity though, is still going strong but has almost no ties to alchemy. Where it gets more complex is when we look at Hermeticism. It still exists but is a very secretive religion (similar to when it was practiced by alchemists). Hermetic people today though are often seen as crazy because they still believe things that have been disproved for centuries.
Even though the accuracy of alchemy is questionable, no one can deny that its stories are incredible. "A search for magical artifacts and substances, shrouded in mystery, outlawed by the powerful and based on an extinct race" sounds like an fascinating idea for a story. Alchemical figures, objects, and ideas can be found in many forms of media. In the book"Harry Potter and the Philosopher's Stone", the villain gets a hold of the philosophers stone and creates the elixir of life. The protagonists seek the aid of Nicholas Flamel to stop him. The Movie "Indiana Jones and the Last Crusade" follows an archaeologist on his hunt for the holy grail. A 2014 Horror movie has the title "As above, So below", a direct quote from the emerald tablet. The Dark Crystal, Full metal Alchemist, Tomb Raider and Assassin's Creed all have elements of the legends in alchemy.
Alchemy was a product of its time. Religion was the only thing people truly trusted and the only way to make discoveries was through religion. With the general belief in religion waning and the accuracy of science becoming ever more clear, alchemy was doomed. Hermeticism was based on science that was proven wrong and as the basis of alchemy, when the religion ended, so did the study.