Learn English – “Correct” way to describe “looking at someone with new eyes” or similar

There are some reasons to believe that this peculiarity of today's English can be ascribed to the influence of Celtic Languages with which English has been in contact for the last 1500 years.

Celtic influence

Here are a few examples:

• In Scottish Gaelic, you ask someone’s name by saying “Dè an t-ainm a tha ort?”, literally “What’s the name that's on you?”.
• In Irish it would be “Cad is ainm duit?” (formal) – “What name do you do?”.
• Or informally, in Irish again, “Cén t-ainm atá ort?”, lit. “What's the name on you?”.
• In Breton (of France) “Petra eo da anv?”, lit. “What is your name?”.
• Cornish “Puth yw dha hanow?”, “What is your name?”.
• Welsh (formal): “Beth ydych enw chi?”, “What is name you?”.
• Welsh (informal), “Beth ydy dy enw di?”, “What is your name from?”.
• Welsh (informal), “Beth yw dy enw?”, “What is your name?”.

General Celtic Influence

Several linguists including J R R Tolkien have claimed that there is a British Isles Sprachbund incorporating English and the Celtic languages. For instance the “I am working”, “I was working” is common in Welsh and Irish, but you can’t say “Ich bin arbeitend” in German.
See the book "The Celtic Roots of English" for more of these.

The Old English way of asking "what's you name" was Germanic

If you had a kind of Rosetta Stone handbook to learn Old English what would you find? Well there are actually a few ones on the net. Here is one. And what do you think the title for the first question is? Sure enough: “Hu hattest þu?”. Literally "How call you" (In German the verb heißen can be both intransitive "Wie heißt du?", or transitive).
So indeed when the Saxons and the Angles landed in the south East of the British Isles they still used the Germanic construction, which other answers and comments have shown, is still in use today in many Germanic languages.
For instance, out of 91 riddles in the 10th century codex Exoniensis, I've counted 13 occurrences of “Saga hwæt ic hatte” (“Say how I[am] called”) and 5 “Frige hwæt ic hatte” (“Guess how I [am] called”).

When did it all change ?

This is trickier but it is supposed to be during Middle English, the period between 1066 and 1450. I'd like to cite two excerpts of the book Beginning Old English (Carole Hough and John Corbett ).

1. The harrying of the north: The Norman French, themselves descendants of Vikings, led by William the Conqueror, over-ran the Anglo-Saxon kingdoms and extended their territory throughout the Danelaw as far as the northern kingdom of Scotland.

   In their vicious land-grab the Normans depopulated whole areas of Northumbria, carrying out an ethnic displacement later called ‘the harrying of the north’. Refugees from the defeated Anglo-Saxon dynasty fled with their retainers and servants to the court of the Celtic-speaking Scots in Edinburgh. There the Anglo-Saxon Princess Margaret married the widower King Malcolm. The speech of Queen Margaret and the Northumbrian refugees would eventually spread out over the Scottish lowlands, and become the basis of the lowland Scots tongue.

2. And a similar one taking place after the harrying of the north:

   For around two centuries after the establishment of Norman rule in England, English was spoken but relatively seldom written. Even so, the influence of English continued to spread. Although Scotland was a separate kingdom, King David, the heir of Malcolm and Margaret, established peaceful relations with many powerful Norman barons, granting them land in the Scottish lowlands. These barons brought with them many English-speaking retainers, mainly from northern England, where there was a strong Norse influence. A distinct variety of the language, first known as ‘Inglis’ and much later as ‘Scottis’, evolved. Today, the pronunciation and vocabulary of the lowland Scots language is often very close to its Old English (OE) origins, as in ‘hoose’, ‘moose’ and ‘coo’ (OE hus, mus, cu)

This is consistent with a change occurring in Middle English. I find other explanations of the Celtic substrate re-emerging less convincing because of the 5 century gap between the invasion of the Anglo Saxons and the Exeter Book.

Online Resources

• "What is your name?" in more than 330 languages”.
• Wiktionary for all the small words included in the Celtic part.
• Brittonicisms in English.
• Middle English creolisation.

"The ball is in your court", "It's time to decide/choose.", "It's time to make your decision/choice", "It's up to you.",

In any case, I'd follow with a pregnant pause, heh.