It's very difficult to give you any specific advice on this without really understanding what it is you're trying to achieve. To get to destination B you have to have a starting point A!
What are you trying to achieve here?
Have you said, hey I've got this bottle of Almond Milk I have to use up, perhaps I'll try and make a dipping sauce with it? That's a fun way to do things (I do it a lot) or are you saying, hey I really like the subtle flavour of Almonds, I'd really like to try and make a dipping sauce with that flavour in it?
Almond milk in itself suggests more of a coating sauce than a dipping sauce, along the lines of the foundation sauces I would perhaps experiment with starting with a roux (fat and flour) and creating a sauce with your almond milk ala Bechamel.
Now you've clarified what you're trying to achieve, I think I would definitely try the roux method first. As I said, I think there are two ways to go here, using a roux as a base and creating an almond butter.
You'll have to do a fair bit of experimentation here, but isn't that half the fun of cooking in the first place? I find that with even established printed recipes, I have to cook it a few times before I fully understand what's going on, so expect to make a few runs though tweaking ingredient amounts etc before you're fully satisfied.
Start by making a basic roux based sauce using the almond milk. The basic ratio for this is 1 cup milk to 1 tablespoon butter (or oil or even almond butter) to 1 tablespoon of flour. Again you could experiment with these ingredients - try almond butter or part almond butter with butter. Try normal flour, or part almond flour etc. You may need to tweak the quantities for your own tastes, less roux and more milk for a thinner sauce, more roux, less milk for a thicker sauce.
The beauty of this is you can tweak the ingredients to acheive the desired thickness of the sauce almost from a runny coating sauce to a thicker dipping sauce.
You could try simmering the sauce with other ingredients to tweak the colour/flavour - saffron, cloves, cinnamon, nutmeg, sugar, for a sweeter sauce or peppercorns, bay leaf, pinch of salt and white pepper for a more savoury one.
Don't be afraid to experiment and try out different things till you achieve your objective, cooking is meant to be fun!
Best Answer
Your best bet for this is xanthan gum, which is an excellent stabiliser. Whilst the distinction between 'natural' and 'unnatural' is fraught with difficulties, insofar as xanthan gum is a product of microbial fermentation then it is no more 'unnatural' than alcohol or vinegar.
Be careful not to use too much though (unless, of course, you want your drink to have the texture of mucus), you really only want to use about 0.4% of the volume of the liquid you're trying to stabilise: so if you have 500ml of milk then you would use 2g of gum. Add it to your liquid and use a blender or an immersion stick to incorporate, a few pulses should do, any more than that and you'll just incorporate air and you could just end up with an annoyingly stable foam!