Jane won the lottery.
(a) I suggested that she buy a car with the prize money.
(b) I suggested that she should buy a car.
(c) I suggested buying a car.
(d) I suggested that she bought a car.
All of the above are acceptable forms in English. Although a careful American speaker–unfamiliar with British English–might find (d) awkward and view it as incorrect.
The verb "suggested" is in the past, so the suggestion was made at a specific point in time. In (a), (b) and (c) the winner of the lottery may or may not have decided to purchase a car. Buying a car was an idea, something for her to consider, but we don't know what happened after that.
Without further context, (d) is ambiguous.
(d) I suggested that she bought a car.
This could mean that the winner of the lottery did in fact buy a car after the idea had been put forward. It was my idea for her to buy it.
OR
The winner did not buy a car; she ignored the suggestion and spent the money she'd won on something else.
For reference see the website English Grammar
When the verb in the main clause is in the present tense, we can use a
present tense in the that-clause after suggest. When the verb in the
main clause is in the past tense, a past tense is possible in the
that-clause as well.
- His doctor suggests that he stops smoking.
- His doctor suggested that he stopped smoking.
In American English, subjunctive structures are
more common after suggest. Note that a subjunctive has the same form
in the present tense and the past tense. It doesn’t have the –s
marking in third person singular.
- The doctor suggests that he stop smoking. (Subjunctive with no –s
marking.)
- The doctor suggested that he stop smoking.
In British English, subjunctive isn’t very common. Instead, British speakers use should + infinitive.
- The doctor suggests that he should stop smoking.
- The doctor suggested that he should stop smoking.
In direct suggestions that begin with I suggest…, should is not
normally used.
- I suggest that you get another job. (NOT I suggest that you should get another job.)
Alternatively, the following is also acceptable
- I suggested that you got another job.
Using the past tense of "got" here appears to be more common in BrEng, implying (perhaps) that the person did the thing suggested. However, without context this is, at best, a supposition.
From the BBC site (which is British) we have the following examples of past form of the verbs in the that-clause
- I insisted / demanded that he stopped phoning me.
- Everybody recommended that she continued with her education for 3 more years.
- It was desirable (that) she went to a school where Italian was taught.