The building completed in 1888 in Mumbai that famously fuses Victorian Gothic Revival architecture with traditional Indian design elements is the Chhatrapati Shivaji Maharaj Terminus (CSMT), originally named Victoria Terminus (VT) after Queen Victoria.

Key facts that make it the answer:

• It was designed by the British architect Frederick William Stevens and took about ten years to build, opening in 1888 — matching the date in the question exactly.
• Its style is a distinctive blend: the Gothic vocabulary of pointed arches, ribbed vaults, gargoyles, turrets and stained glass is combined with motifs drawn from Indian palace architecture, giving it a uniquely Indo-European character.
• It was inscribed as a UNESCO World Heritage Site in 2004 and remains the headquarters of the Central Railway and one of India's busiest stations.

The other options do not fit the 1888 date or the described style:

• Bombay Town Hall was completed much earlier (1830s) in a Neoclassical style with Greek/Roman columns — not Gothic.
• Gateway of India was completed in the 1920s in the Indo-Saracenic style, and commemorates a royal visit rather than serving as a railway terminus.
• Rajabai Clock Tower (around 1878), though also Gothic and by the same era's designers, is a university clock tower and does not match the 1888 completion or the specific description given.