I haven't found a consensus for the definition of an imaginary cuil, therefore i offer this interpretation.

An imaginary cuil may represent a contradiction in a given scenario. i‽ cuil represents one contradiction. for example:

You ask your waiter for a hamburger…

0‽:Your waiter hands you a hamburger.

i‽:Your waiter without hands, hands you a hamburger.

2i‽:Your mute waiter without hands, hands you a hamburger and says, "Thank you for your patronage."

It's important to note that although these might seem to be abstractions they don't pose any deviation from expectation as you still receive a hamburger from your waiter. Although contradictions might be considered abstractions by some people, these can't be some form of abstraction as it is impossible to conceive of their occurring. A mute by definition can not say anything, and a hand-less person can not by definition hand you something (we will assume for our purposes that handing something entails giving with hands). In examples of real number cuils there have never been contradictions of definition. All scenarios have been possible on a theoretical level even if they were impossible on a substantial level. an imaginary cuil represents any theoretical impossibility.

If a scenario entails some abstraction along with a contradiction then it can be represented as having a complex value for a cuil.

for instance:

You ask your waiter for a hamburger…

(i+1)‽:Your waiter without hands, hands you a raccoon.

a non integer imaginary value would represent a scenario in which there are multiple contradictions but there is an overlap in the contradictory definitions

for example:

You ask your waiter for a hamburger…

1.5‽:Your waiter without hands, hands you a hamburger and gives you a high-five.

The act of handing someone something contradicts what it means to be without hands, and the act of high-fiving someone also contradicts this. this is represented by one and one half i ‽.

and finally any negative imaginary cuil would represent a specification that renders contradiction impossible.

for example:

You ask your waiter for a hamburger…

-i‽:your waiter, who has a hamburger, with his hands, hands you a hamburger.

this added specifications insures that there is no possible contradiction from the word hands (as a verb).

Please let me know what you think of this interpretation of imaginary cuil, Tell me your own interpretations, and tell me if there is already a standardized interpretation of imaginary cuil.