I can't imagine having another dog when I think about how my dog died.
When I think about the time when my dog died, I can't think of owning another dog.
ご質問ありがとうございます。
①
"I can't imagine having another dog"＝「他の犬を持つことを想像できない」
"when I think about how my dog died."＝「自分の犬が死んだときのことを考えると。」
②
"When I think about the time when my dog died"＝「私の犬が死んだときのことを考えると」
"I can't think of owning another dog."＝「他の犬を持つことが考えられない。」
ご参考に！
アメリカ合衆国 
