I'm not sure 'imaginary' is the right word. Value is real whether or not it's a social construction.
There's a widespread misperception that money is some physical entity. It is not.
A description I've come to use is that money is the medium of greatest acceptance within a given market or region. Note that money need not be paper notes, coins, or even any sort of government issue. There's an excellent 1945 paper on the economic organisation of a POW camp which describes how an economy based largely on cigarettes and various Red Cross ration items emerged amongst Allied prisoners of Germany during WWII:
<https://www.jstor.org/stable/2550133>
In particular, it looks at what problems money can solve (and what happens when there's an insufficient money supply --- in this case, cigarettes), as well as those it cannot (an insufficiency of goods generally to transact).
At various points in time, clams, beaver pelts, cowhides, knives, massive multi-tonne stones (<https://www.npr.org/sections/money/2011/02/15/131934618/the-...>), letters of credit, and cryptographic hashes have served as money. In mediaeval Europe, Roman coins were long used in trade, well after the Roman empire itself had fallen. In parts of the world, US dollars are a preferred currency even outside the 50 states and US territories, with several countries officially adopting the US dollar as their own national currency: <https://www.investopedia.com/articles/forex/040915/countries...>.
William Stanley Jevons defined what to him were the vital set of properties required of money in Money and the Mechanism of Exchange (1877): utility/value, portability, indestructibility, homogeneity, divisibility, stability, cognixability.
<https://archive.org/details/moneyexchange00jevorich/page/30/...>
I disagree with him on the first property. Money may have an intrinsic value (as with gold or specie), but need not. In particular, the intrinsic value of money is inverse to the trust in the monetary authority itself. That is, in a low trust financial system, money typically consists of or is backed by some physical store of value. In a high trust financial system, currency tokens need have no fundamental utility (as with paper banknotes or digital accounts), but rather there is a trust in the system as a whole to function predictably and reliably.
The value represented by money is one that is socially, legally, and economically recognised. It's ultimately a tokenisation of credit and wealth. It is not directly tied to any physical characteristics (though as a medium of exchange it can be traded for any given physical commodity or service). That's not "imaginary" in the same sense that other social conventions such as which side of the road to drive on are not imaginary. Which side to drive on is entirely arbitrary as an initial social choice, but once that side has been chosen there are very real consequences to flouting the convention.
Social construct != imaginary.
