The only country with a coastline on both the Caspian Sea and the Indian Ocean is Iran.
Iran reaches the Caspian Sea along its northern edge and the Gulf of Oman along its south-eastern coast. The Gulf of Oman is a marginal sea of the Indian Ocean, connecting the Arabian Sea with the Strait of Hormuz. Iran also borders the Persian Gulf, which is connected to the Gulf of Oman through that strait.
This gives Iran a remarkable geographic span: its northern Caspian shore is separated from its southern maritime coast by the Alborz and Zagros mountain systems and the broad Iranian Plateau. The country has about 740 kilometres of Caspian shoreline, while its southern coasts face the Persian Gulf and Gulf of Oman.
A common mix-up is Pakistan, which has an Indian Ocean coastline but no Caspian coast, and Azerbaijan, which borders the Caspian but not the Indian Ocean. Russia also touches the Caspian Sea, yet its other coasts face the Atlantic, Pacific, and Arctic oceans rather than the Indian Ocean.