Necessary? No, but...
No, it is not necessary; a class will certainly function and be playable without any archetypes. For any one character, it won’t really even be noticeable.
The purpose of archetypes is to give a clear way in which different members of a class are, in fact, different from one another. They also provide an easy route to expansion—if there is already a defined archetype “slot,” and you think of some new, related ideas, you don’t need to create a whole new class from scratch, you can just create an archetype.
Leave yourself an opening
My suggestion is, if nothing else, leave yourself an opening here by creating an archetype “slot”—say some of the features are from an archetype. So your class has archetypes, it just has only one at the moment. That means you can create more later. That gives you the second benefit of archetypes, and that benefit lets you get the first benefit more easily as the class gets played more.
Alternatively, maybe you should widen your scope
Maybe, if you can’t think of archetypes for an alchemist class, you should think of alchemist itself as an archetype of a broader class. This is actually what Wizards of the Coast did—they have “alchemist” as an archetype of their “artificer” class.
Real-world alchemy was pretty varied
Just for the sake of ideas, the history of alchemy in the real world is pretty interesting and has a lot of variety. For example, some alchemists were much more mystics and philosophers, barely dealing with physical substances at all (a spellcaster archetype maybe?), while others were really proto-chemists, with barely any mysticism (an archetype with less supernatural elements, but more reliability, perhaps?). Still others weren’t either of those things—they were just charlatans, conning some lord. That could be (along with some default middle-of-the-road case) four archetypes, perhaps.