In "The Book of Sand" (1975), Jorge Luis Borges describes a volume of inconceivably thin leaves in which no page is the first and no page the last, so that wherever you open it there is a different story, written in various indecipherable scripts. The narrator becomes obsessed with this extraordinary object and ultimately horrified: "I realised that the book was monstrous. It was no consolation to think that I … was no less monstrous than the book."

The short story echoes what is probably Borges's single most famous fiction, "The Library of Babel" (1941), which depicts a library of astronomical size containing everything that ever has been or could be written but in which meaning is elusive. The later work, however, written towards the end of the author's life, has a nightmarish quality that is less apparent in the earlier story.

Falling between these two is The Book of Imaginary Beings, a compendium of brief, almost stark descriptions and stories about fantastic animals from many older texts and sources, including the bestiaries of medieval Europe and their classical antecedents, Chinese and Indian myth, folk tales, the legends of indigenous peoples, and the minds of writers such as Kafka and Poe. First published in 1957, at the very time when (as Borges later explained) the vision that had gradually been failing him since birth had deteriorated to the point where he could no longer read or see what he was writing, this cryptozoological chiaroscuro is one of Borges's great creations.

In the preface, Borges warns that Imaginary Beings is not meant to be read straight through: "Rather, we should like the reader to dip into these pages at random, just as one plays with the shifting patterns of a kaleidoscope." This is good advice. Though brief – less than 160 pages – Imaginary Beings is dense and deep. I have read it several times over the years and am always coming across new things in it.

Many of the more than 100 entries are delightful – and amusing – from start to finish. The beasts include the Upland Trout, which nests in trees and is a good flier but scared of water, and the Goofang, which swims backwards to keep the water out of its eyes. Others are simply weird. What are we to make of the Strong Toad, which has a shell like that of a turtle, glows like a firefly in the dark, and is so tough that the only way to kill it is to reduce it to ashes? "Fauna of Mirrors", begins with what looks like a joke on erudition of Shandy-esque proportions: "In one of the volumes of the Lettres édifiantes et curieuses that appeared in Paris during the first half of the 18th century, Father Fontecchio of the Society of Jesus planned a study of the superstitions and misinformation of the common people of Canton … "

Imaginary Beings is also populated with creatures that, however bizarre, are far from absurd. Notable are the various dragons of east and west – creatures of enormous power but uncertain significance. "We are ignorant of the meaning of the dragon as we are ignorant of the meaning of the universe," Borges writes in his preface, "but there is something in the dragon's image that appeals to the human imagination … It is … a necessary monster."

The book is, then, something like a map of the endless labyrinth of human imagination and its contents. It holds, as it were, a mirror up to dreaming. But where do pleasant dreams shade into nightmares, or those from the past into those of the future? Which dreams are wholly fantastical and which are visions or distortions of what is real or has the potential to be so?

Such questions might not loom so directly were it not for the magnetic pull of the writing. The jovial opening to "Fauna of Mirrors" unfolds into a story barely a page long that resonates as profoundly as many first-rate sci-fi novels. At his best Borges is, like Dante, a master of the moment as a cypher of a life. He can, like Elizabeth Bishop in remarkable poems such as "The Man-Moth", create in just a few lines worlds that are fully imagined but only partly revealed.

"Fauna of Mirrors" also foreshadows one of a series of lectures given in 1977 in which Borges describes his recurring nightmares (which, like much of his fictional output, feature labyrinths and mirrors). In the most terrible of all, he sees himself reflected in a mirror but the reflection is wearing a mask such as he had feared greatly in childhood. "I am afraid to pull the mask off, afraid to see my real face, which I imagine to be hideous. There may be leprosy or evil or something more terrible than anything I am capable of imagining."

But The Book of Imaginary Beings can be viewed in other ways than as a commentary on re-enactment of human dreams. It can, for example, remind us of what is beyond dream – the real forms of living creatures that exist without human agency. Borges himself acknowledged as much in his preface: "Anyone looking into the pages of the present handbook will soon find out that the zoology of dreams is far poorer than the zoology of the Maker."

For we who live in the light of what paleontology, evolutionary biology and genetics are revealing about living forms, our response to the real may – will, if we are truly awake – be one of astonishment and wonder at life's inventiveness. Even ordinary-seeming animals are marvellous in the light of evolution: the chicken, for example, is the closest living relative of Tyrannosaurus rex. Extraordinary ones make those in the pages of a medieval bestiary seem poor indeed. Compared to the leafy sea dragon (pictured, a cousin of the seahorse that looks very much like seaweed and yet also like a dragon) and the sea slug Elysia chlorotica (which photosynthesises with genes stolen from the algae it eats, and is as green as a leaf), the mythical Barometz, or vegetable lamb of Tartary, is a dull affair.

The contemplation of natural history allows us to marvel at our place in the universe. As Charles Darwin wrote early in his career, "If, as the poets say, life is a dream, I am sure in a voyage these are the visions which serve best to pass away the long night."

Yet another way of reading The Book of Imaginary Beings is in the shadow of the future – as phantasmagoric counterpart to the Anthropocene, the epoch in which humans may be transforming life as radically as anything since the Cambrian explosion about 530m years ago. A global rise in average temperature of 4C or more by the end of this century, which many scientists now consider quite likely, will lead to a disruption of the biosphere. One of the biggest extinction events in the history of life may be unfolding. At the same time, we are on the verge of creating new forms of non-human life, if we are not already doing so, as well as new ways of being human (children with DNA from three parents, reproduction without the union of male and female gametes, greatly extended lifespans). Such endeavours may open up countless new ways to imagine and to be.

Borges anticipates manipulation of life in an entry on the golem. The Kabbalists sought to rearrange the letters of the ineffable names of God in their attempts to make new life. Today, we are tinkering with the near endless potential variations in the letters of DNA. How hopeful will the resulting monsters be? How much light or darkness will they bring? "The light which puts out our eyes," wrote Thoreau, "is darkness to us."

The Book of Imaginary Beings tosses stone after stone into the subterranean caverns of the reader's mind. It takes us along passageways and turns corners to reveal strange shapes and images, some of which may precede and outlast anything conceived by man. If we are attentive, the reverberations can help us trace the dimensions of those spaces. We glimpse chinks of light and are then engulfed in sudden dazzling floods of it.

• Caspar Henderson's The Book of Barely Imagined Beings: A 21st Century Bestiary is published by Granta.