The basic forms are:
Imagine if + [present tense clause].
Imagine if + [past tense clause].
Imagine if + [past participial clause].
The pattern is the same as in conditionals. There can be some ambiguity because some words have the same past and present tense. This is the case with "put".
Without context, we do not know whether "Imagine if they put me ..." is 1 or 2.
A present tense clause refers to an imaginary future.
Imagine if the interest rate doubles; what will that do to our mortgage payments? [A concern about the future: imagine a future in which the interest is double.]
A past participial clause refers to an imagined past which did not happen, which would change the present:
Imagine if you had been seriously hurt in your accident; where would you be today? [You were not hurt in your accident, but imagine what the world would be now if that happened.]
A past clause is ambiguous; it can refer to an imagined past leading to an imagined present, or to imagined future events. In informal speech, a simple past is used in conditionals instead of the past participle.
Imagine if someone stole your car. [Your car was not stolen, but imagine a present world, in whose past your car was stolen. Or: your car has not been stolen, but imagine a future in which someone steals your car. ]
Imagine if you had lots of money. [Imagine a world in which you have lots of money.]
Imagine if your parents sent you to boarding school. [Imagine a future in which your parents have sent you to boarding school. Or, imagine a present in which you have been sent to boarding school.]
The second example above isn't ambiguous, but the first and third are.
Additional clauses clear up the ambiguity:
Imagine if someone stole your car. What would you be doing now? [Past -> Present]
Imagine if someone stole your car. What would you do? [Future]
If you want to be clear and precise in writing, use a participle form, or a present tense:
Imagine {if someone had|that someone has} stolen your car. What would you be doing now?
Imagine {if|that} someone steals your car. What will you do?
