John H. Thompson was an American politician who served as the fourth Lieutenant Governor of Indiana from 1824 to 1828.
From 1824 to 1828, Thompson served as Lieutenant Governor under William Hendricks and James B. Ray.
His candidacy was supported by Jacksonian Democrats but he lost the election to former Governor Jonathan Jennings.
In 1830, Thompson ran as an independent for the same seat, joining a crowded field that also included Jennings, John Carr, William W. Wick, James B. Ray, and Isaac Howk.
In 1839, Thompson ran as a Democrat against fellow Democrat Johnathan McCarty and Whig James Rariden for the U.S. House seat of Indiana's 5th congressional district.