首先look like(看起来像)为表象系动词(look)+介词(like)结构, like的词性为介词，不是连词。

The beautiful mausoleum was designed to look like what the emperor hoped would be Mumtaz's home in the afterlife.

参考译文:这座美丽的陵墓被设计成皇帝希望将会成为墨姆塔兹死后的家。

备注(1):what the emperor hoped would be Mumtaz's home in the afterlife为what引导的嵌入式名词性关系分句，作介词like的宾语成分。

备注(2):what 宾语从句里面又嵌套一个主从复合句，the emperor hoped 为主句，what would be Mumtaz’s home in the afterlife为从句，即是谓语动词hoped的宾语从句。what为宾语从句的主语。

备注(3):what是谓语动词hoped 后面宾语从句的主语成分。相当于the emperor hoped what would be Mumtaz’s home in the afterlife.

特别备注:在传统语法里：the emperor hoped被分析为插入语。What为介词like宾语从句的主语。

整个句子语法结构分析如下供网友参考

【1】The beautiful mausoleum【主语】// was designed【谓语动词】//  to look like what the emperor hoped would be Mumtaz's home in the afterlife【主语补足语谓语动词】// .

不定式短语的语法结构分析如下供网友参考

【2】to look【不定式】// like what the emperor hoped would be Mumtaz's home in the afterlife【表语】// .

介词短语的语法结构分析如下供网友参考

【3】like【介词】// what the emperor hoped would be Mumtaz's home in the afterlife【宾语从句】// .

介词宾语从句的语法结构分析如下供网友参考

【4】what the emperor hoped would be Mumtaz's home in the afterlife

=The emperor【主语】//hoped【谓语动词】// what would be Mumtaz's home in the afterlife【宾语从句】// 

主句:The emperor hoped

从句:what would be Mumtaz's home in the afterlife