Multiplexer + flipflop all in one?
More precisely, what you're describing is a decoder, not a multiplexer.

A decoder is a purely combinational circuit. To make it into sequential circuit which holds state, the easiest way to do that seems to be to put a latch in front of it. In your case, a four bit latch.

There are IC's like this:

74HC4514

This is a "4-to-16 line decoder with latch".

As for multiplexers, they do something different. Based on an \$n\$-bit binary number input, one of \$2^n\$ possible signal lines are routed to the output. Whereas driving one of \$2^n\$ lines high (or low) based on a binary input is decoding.