The phrase 'I began questioning' is correct and usable in written English.
You can use it to describe when you started to ask a series of questions. For example, "After what I witnessed, I began questioning the integrity of the company."
Everybody laughed, and I began questioning the disciples.
After this project, I began questioning form in a new, deeper manner: from genre, to sentence, to word.
"During that period when I was really homeless, I began questioning a lot.
None of these discussions could placate the sense of guilt I felt and I began questioning whether I was made to be a social worker.
I began questioning myself.
I began questioning myself, "What was I thinking?
It is only when I begin questioning him about the sadistic horrors that he visits on the audience in Antichrist - a penis spurting blood, a graphic depiction of female genital self-mutilation - that he suddenly sits up straight again and looks uncomfortable.
I'm probably going to start to sound like an old fart when I begin questioning why in the hell we think it might be a good idea to perpetuate this idea of shopping in the middle of the night for something we couldn't possibly live without until 3 the following afternoon.
During this period of time I grew more disillusioned about traditional learning environments and began questioning why students even went to school.
Eventually, I moved beyond feeling gratitude for my opportunities and began questioning why these opportunities don't happen to more people like me.
"It struck me as odd, because I don't see how homeownership had any bearing on being seated in a restaurant," said Ms. Espy-Edwards, who began questioning Mr. Lecendre.
