The question asks us to identify a British colonial building in Mumbai, completed in 1888, which showcases a blend of Victorian Gothic and Indian architectural elements. Let's analyze the options:

1. Chhatrapati Shivaji Maharaj Terminus: This is a historic railway station in Mumbai, designed by British architect Frederick William Stevens. The building is renowned for its Victorian Gothic Revival architecture, interspersed with traditional Indian elements. It was completed in 1888, adhering to the criteria given in the question.
2. Gateway of India: Built in the 20th century and completed in 1924, this monument was constructed to commemorate the visit of King George V and Queen Mary to Mumbai. Though prominent, it does not fit the timeframe or architectural blend described in the question.
3. Bombay Town Hall: Also known as the Asiatic Society of Mumbai, this neoclassical structure was completed much earlier, in the 1830s. It does not reflect the Victorian Gothic architectural style combined with Indian elements.
4. Rajabai Clock Tower: This structure is part of the University of Mumbai and was completed in 1878. While it is an example of Venetian and Gothic styles, it was completed earlier than 1888 and primarily serves a different architectural purpose.

Given this analysis, the correct answer is Chhatrapati Shivaji Maharaj Terminus. It precisely matches the question's criteria of being completed in 1888 and featuring a blend of Victorian Gothic and Indian architectural elements.