Quito

City · 1534 CE · in South America

Capital of Republic of Ecuador (1840–today).

Thanjavur (c. 1534 CE): Capital of Nayaks of Thanjavur (1534–1673), Thanjavur Maratha Kingdom (1677–1799).
La Paz (1535 CE): Capital of Republic of Baja California (1853–1854).
Lima (1535 CE): Founded by Francisco Pizarro as the "City of Kings" in 1535, Lima became the capital of the Viceroyalty of Peru and of independent Peru.
Buenos Aires (1536 CE): First founded by Pedro de Mendoza in 1536 and refounded in 1580 by Juan de Garay; capital of the Viceroyalty of the Río de la Plata and of Argentina.
Asunción (1537 CE): Capital of Paraguay (1811–today).
Bogotá (1538 CE): Capital of Gran Colombia (1822–1831), Republic of New Granada (1834–1858), Granadine Confederation (1859–1862), and 2 more.
Sucre (1538 CE): Capital of Bolivia (1825–today).
Serang (c. 1529 CE): Capital of Sultanate of Banten (1529–1808).
Lahore (c. 1529 CE): Capital of Mughal Empire (1529–1857), Sikh Empire (1799–1848).
Johor Bahru (c. 1529 CE): Capital of Sultanate of Johor (1529–1884).

Around the same time and place

Francisco Pizarro (c. 1478 CE – 1541 CE): Spanish conquistador who led the conquest of the Inca Empire and founded Lima in 1535; he was assassinated there by rival Spaniards.
Battle of Cajamarca (1532 CE): Francisco Pizarro's small Spanish force ambushed Atahualpa's retinue in Cajamarca's plaza and captured the Inca emperor, beginning the fall of the Inca Empire.
Pizarro's march on Cusco (1532–1533 CE): Francisco Pizarro’s small force marched from the coast to Cajamarca, captured the Inca ruler Atahualpa in November 1532 and later executed him, then advanced along the Inca road to occupy Cusco in 1533.
Atahualpa (c. 1502 CE – 1533 CE): Last Sapa Inca, victor of the Inca Civil War; captured by Pizarro at Cajamarca, he paid a huge ransom in gold and silver but was executed in 1533.
Balboa sees the Pacific (1513 CE): Vasco Núñez de Balboa led an expedition across the Isthmus of Panama and in September 1513 became the first European to see the Pacific Ocean from the Americas, claiming it for Castile.
Panama City (1519 CE): Capital of Republic of Panama (1905–today).
Muisca (c. 600–1537 CE): Chibcha-speaking people of the Colombian highlands renowned for goldwork; their raft ceremony at Lake Guatavita inspired the legend of El Dorado.
Columbus’s fourth voyage (1502–1504 CE): On his last voyage Columbus explored the Caribbean coast of Central America from Honduras to Panama, then was stranded for a year on Jamaica after his worm-eaten ships failed.