The answer will come down to what flavour you want to get out of it. Cardamom is cooked in water (ground and added to Arabic coffee, or whole in some rice dishes, e.g. some forms of Pulao). On the other hand it's also crushed and cooked in oil in many spice blends including some Tadka recipes.
So now that we've established that both are real options, how would you start to make a decision?
The main flavour compounds as follows, from a list on Wikipedia. I've included the melting points because compounds that are liquid at cooking temperatures don't need to be dissolved to be extracted. Some will get out even if they're not soluble or even miscible; oils can sometimes be seen on the surface of hot water-based drinks for that reason.
Of course some compounds have stronger flavours than others, so measured solubility alone isn't a good guide to the flavour that will be imparted to a solution. But it points us in the right direction.
So many of these compounds that contribute to the flavour will be barely extracted at all in (hot) water, and oil is needed for the full profile. However simmering or steeping in hot water is perfectly valid.
It proves easier to define solubility than "flavour of cardamom", as the latter is made up of many compounds (most of which are also present in other herbs and spices.
Returning to "dump that medium into whatever I'm cooking" from the question, if you do this with spices cooked in oil, that's basically a tadka (the spices are added along with the oil). Using cardamom-infused oil to substitute fat in a recipe could work, but cardamom-infused water would give a partial, subtle flavour at most. That's fine if it's what you want in a wet recipe, but dry recipes like these cardamom biscuits add it in ground form for a reason - there's no oil or water to act as a carrier (you could infuse the butter, melted, but that's a lot of hassle for a little flavour)
Most links Wikipedia, most values from there or other links on the same table row