In a memo to staff on Monday, Thompson wrote: "Christiane has thought carefully about this decision, and she and I have been discussing it for some months.
"Like so many CNNers who will hear the news today, I'm very sorry to see her go, but I've come to accept that this is the right time for her, both personally and professionally.
"In the meantime, on behalf of the entire CNN leadership team, I want to thank Christiane for 43 extraordinary years of remarkable journalism, loyalty and friendship," he added.
In her own statement, Amanpour said she wanted to thank "all the amazingly talented people I've worked with throughout this incredible journey. Especially my beloved show teams and field crews."
Amanpour reflected on her "very first big exciting assignment covering the First Gulf War, to the horrors of the Bosnian War, Rwanda, Afghanistan, Ukraine and Iran and all the points in between".
"It's the teams and the camaraderie that counted so much," she continued, "as we together battled to seek the facts, the evidence and the humanity, so that CNN could always broadcast the truth about the world and to the world, without fear nor favour."