user20721: What is the name of the chess piece in a variant game where there is a piece that moves like the queen AND a knight?
Is there a name for this variant?

user3819: The Chess variants site has introduced tags recently and you can find some games with the tag Piece:Amazon. As of the day of writing this answer, the tagging is not yet comprehensive and a lot of games featuring the Amazon are still missed by this but I expect things to improve over time.
In addition, there is a list of games with the Amazon in the Piececlopedia entry Amazon on the same site.

user19877: According to the Wikipedia variants page, there does not seem to be a variant with a specific name, but there are several variants that use the queen-knight hybrid, which is called an Amazon (appropriate name, and combines rook+bishop+knight movement abilities). So maybe just "Amazon chess" would make sense.
The Amazon is used in the following variants (copied from the Wikipedia page above):

Maharajah and the Sepoys: Black has a complete army, and White only one piece: the maharajah (Amazon).
Musketeer Chess: A commercial variant, inspired from Seirawan Chess. This variant introduces 10 fairy pieces: archbishop, chancellor, hawk (different rules from Seirawan Chess), elephant (different rules from Seirawan Chess), leopard, cannon (different from Xiangqi), unicorn, fortress, spider, and amazon (also called dragon in this game). Players have a choice of 2 pieces among the 10 possible and method used to introduce them during the game.
Wolf Chess: On an 8×10 board, with fairy pieces wolf (empress), fox (princess), nightrider, sergeant (almost a Berolina pawn), and elephant (amazon). By Arno von Wilpert (1943).

There are also the following hybrid pieces:

Empress (rook+knight)
Princess (knight+bishop)
The traditional Queen (rook+bishop)


20721: <p>What is the name of the chess piece in a variant game where there is a piece that moves like the queen AND a knight?</p>

<p>Is there a name for this variant?</p>
 3819: <p>The Chess variants site has introduced tags recently and you can find some games with the tag <a href="https://www.chessvariants.com/tag/Piece%3AAmazon" rel="nofollow noreferrer">Piece:Amazon</a>. As of the day of writing this answer, the tagging is not yet comprehensive and a lot of games featuring the Amazon are still missed by this but I expect things to improve over time.</p>
<p>In addition, there is a list of games with the Amazon in the <a href="https://www.chessvariants.com/piececlopedia.dir/amazon.html" rel="nofollow noreferrer">Piececlopedia entry Amazon</a> on the same site.</p>
 19877: <p>According to the <a href="https://en.wikipedia.org/wiki/List_of_chess_variants" rel="nofollow noreferrer">Wikipedia variants page</a>, there does not seem to be a variant with a specific name, but there are several variants that use the queen-knight hybrid, which is <a href="https://en.wikipedia.org/wiki/Amazon_(chess)" rel="nofollow noreferrer">called an Amazon</a> (appropriate name, and combines rook+bishop+knight movement abilities). So maybe just "Amazon chess" would make sense.</p>

<p>The Amazon is used in the following variants (copied from the Wikipedia page above):</p>

<ol>
<li><strong>Maharajah and the Sepoys</strong>: Black has a complete army, and White only one piece: the maharajah (Amazon).</li>
<li><strong>Musketeer Chess</strong>: A commercial variant, inspired from Seirawan Chess. This variant introduces 10 fairy pieces: archbishop, chancellor, hawk (different rules from Seirawan Chess), elephant (different rules from Seirawan Chess), leopard, cannon (different from Xiangqi), unicorn, fortress, spider, and amazon (also called dragon in this game). Players have a choice of 2 pieces among the 10 possible and method used to introduce them during the game.</li>
<li><strong>Wolf Chess</strong>: On an 8×10 board, with fairy pieces wolf (empress), fox (princess), nightrider, sergeant (almost a Berolina pawn), and elephant (amazon). By Arno von Wilpert (1943).</li>
</ol>

<p>There are also the following hybrid pieces:</p>

<ol>
<li><a href="https://en.wikipedia.org/wiki/Empress_(chess)" rel="nofollow noreferrer">Empress (rook+knight)</a></li>
<li><a href="https://en.wikipedia.org/wiki/Princess_(chess)" rel="nofollow noreferrer">Princess (knight+bishop)</a></li>
<li>The traditional Queen (rook+bishop)</li>
</ol>
